Drawing Maps with Advice

Size: px
Start display at page:

Download "Drawing Maps with Advice"

Transcription

1 Journal of Parallel and Distributed Computing 7() (0) 4 Drawing Maps with Advice Dariusz Dereniowski Andrzej Pelc Abstract We study the problem of the amount of information required to draw a complete or a partial map of a graph with unlabeled nodes and arbitrarily labeled ports. A mobile agent, starting at any node of an unknown connected graph and walking in it, has to accomplish one of the following tasks: draw a complete map of the graph, i.e., find an isomorphic copy of it including port numbering, or draw a partial map, i.e., a spanning tree, again with port numbering. The agent executes a deterministic algorithm and cannot mark visited nodes in any way. None of these map drawing tasks is feasible without any additional information, unless the graph is a tree. Hence we investigate the minimum number of bits of information (minimum size of advice) that has to be given to the agent to complete these tasks. It turns out that this minimum size of advice depends on the number n of nodes or the number m of edges of the graph, and on a crucial parameter µ, called the multiplicity of the graph, which measures the number of nodes that have an identical view of the graph. We give bounds on the minimum size of advice for both above tasks. For µ = our bounds are asymptotically tight for both tasks and show that the minimum size of advice is very small. For µ > the minimum size of advice increases abruptly. In this case our bounds are asymptotically tight for topology recognition and asymptotically almost tight for spanning tree construction. Keywords: algorithm, advice, graph, topology recognition, spanning tree Introduction. The background and the problem Knowing the topology of a network or at least a spanning tree of it is of significant help in organizing communication among nodes of the network and in accomplishing distributed tasks. It is well known that such tasks as, e.g., broadcasting or gossiping (information exchange) can be performed more efficiently when the topology of the network or a spanning tree of it are available to nodes, than when they have to be executed in an unknown network. One way of supplying this vital information to nodes is by means of exploration of the network, where a mobile agent has to traverse all links of the network and visit all its nodes. In fact, one of the main reasons to perform the well-studied task of graph exploration, see, e.g., [, 4, 5, 4, 5], is to draw a faithful map of the graph that models the explored network. Provided that the agent has enough memory and computation power and that the exploration has been performed, drawing a map of the graph is easy, if nodes have distinct identities that can be perceived by the agent, or if the agent can leave marks at nodes when it visits them. However, neither of these assumptions is always satisfied. On the one hand, nodes may refuse to reveal their identities, e.g., for security reasons, or limited sensory capabilities of the agent may prevent it from perceiving these identities; on the other hand, nodes may not have facilities (whiteboards) allowing to leave marks, or such marks may be destructed between visits of the agent and thus unreliable. Thus it is important for the agent to be able to recognize the topology of the graph (draw a full map of the network) or construct A preliminary version of this paper appeared in the Proc. 4th International Symposium on Distributed Computing (DISC 00), LNCS 64, 8-4. Gdansk University of Technology, ETI Faculty, Department of Algorithms and System Modeling, ul. Narutowicza /, 80- Gdańsk, Poland, deren@eti.pg.gda.pl This work was done during the visit of Dariusz Dereniowski at the Research Chair in Distributed Computing of the Université du Québec en Outaouais. This author was partially supported by the MNiSW grant N N Département d informatique, Université du Québec en Outaouais, Gatineau, Québec J8X X7, Canada, pelc@uqo.ca Partially supported by NSERC discovery grant and by the Research Chair in Distributed Computing at the Université du Québec en Outaouais.

2 its spanning tree (which is an important partial map of it) without relying on identities of the nodes and without marking them. In contrast, in order to allow the agent to move in the network, we have to assume that ports at every node are distinguishable for the agent. If an agent were unable to locally distinguish ports at a node, it may have even been unable to visit all neighbors of a node of degree at least. Indeed, after visiting the second neighbor, the agent cannot distinguish the port leading to the first visited neighbor from the port leading to the unvisited one. Thus an adversary may always force an agent to avoid all but two edges incident to such a node, thus effectively precluding exploration. Hence we assume that a node of degree d has ports,..., d corresponding to the incident edges. Ports at each node can be perceived by an agent visiting this node, but there is no coherence assumed between port labelings at different nodes. We assume that when an agent reaches a node, it learns the port number by which it entered it. We consider two map drawing tasks that have to be accomplished by a mobile agent executing a deterministic algorithm and walking in an unknown connected graph with unlabeled nodes and labeled ports. One is topology recognition which consists in returning an isomorphic copy of the graph with correctly numbered ports, and the other is spanning tree construction which consists in returning a spanning tree of the graph, again with correctly numbered ports. Both tasks have to be performed by an agent that starts in an arbitrary node of an unknown graph and is allowed to explore it. These tasks can be easily accomplished after exploration if the graph is a tree (and in this case they are obviously equivalent): after performing an Eulerian tour of the tree, the agent realizes this fact and can reconstruct the topology of the tree. (By an Eulerian tour we mean a closed walk of minimum length that visits each edge at least once. Note that for a tree, each edge will be visited exactly twice in such a tour.) However, it turns out that unless the graph is a tree, none of these tasks can be accomplished without any additional information given to the agent. (a) (b) Figure : Non-isomorphic graphs undistinguishable by the agent Fig. (cf. [5]) gives an example of two non-isomorphic graphs after whose exploration an agent will get identical information, and thus will not be able to distinguish them. Indeed, a stronger fact is true: for any graph that is not a tree, an agent that has explored the graph and has no additional information can neither recognize the topology of the graph, nor even construct its spanning tree. This yields the main problem that we study in this paper. What is the minimum number of bits of a priori information required by an agent exploring a graph, in order to recognize its topology or to construct its spanning tree? This way of stating the problem follows the paradigm of network algorithms with advice that has become recently popular (cf. the subsection Related work ), and can be described as follows. An oracle knowing the entire network can give a string of bits (advice) to the mobile agent. (In other settings strings of bits are given to nodes that subsequently exchange messages.) Then an algorithm is executed by the agent without knowing in which network it operates, using the provided advice. The total number of bits given by the oracle is the size of advice. Thus the framework of advice permits to quantify the amount of information needed to solve a network problem, regardless of the type of information that is provided. Since for trees no additional information is needed, in the rest of the paper we assume that the explored graph is not a tree. It turns out that the size of advice needed to solve the two problems under investigation depends on three parameters: the number n of nodes, the number m of edges, and a crucial parameter µ called the multiplicity of the graph. Our multiplicity parameter µ is related to the symmetricity σ of a graph, defined in [5]. We define multiplicity for a graph G with a particular port labeling, while σ(g) is defined in [5] to be the maximum multiplicity over all port labelings of G. This parameter depends on the notion of the view from a node. Intuitively, this is the infinite tree with labeled ports, rooted at the given node, that would be obtained by a complete infinite exploration

3 of the graph, if each visited node were attached as a new node in the tree (see the formal definition in Section ). Since nodes of the graph are not labeled and cannot be marked, and thus already visited nodes cannot be recognized on subsequent visits, the view from a node is the maximum information that can be obtained by an agent starting at this node and exploring the graph (cf. Proposition.). It has been proved in [] that if views of two nodes of a n-node graph are different, then their views truncated to level n are also different. Hence, knowing any upper bound on n, the agent can learn in finite time which nodes have equal views and which do not. It is known (cf. [5]) that, for every node v of a graph, the number of nodes from which the view is identical as that from v is the same. This number is the multiplicity µ of the graph.. Our results We give bounds on the minimum size of advice required by an agent both for the task of topology recognition and of spanning tree construction. (We focus on the feasibility of accomplishing these tasks and not on the complexity of their performance.) For µ = our bounds are asymptotically tight for both tasks and show that the minimum size of advice is very small: for an arbitrary function ϕ = ω() it suffices to give ϕ(n) bits of advice to accomplish both tasks for n-node graphs, and Θ() bits are not enough. For µ > the minimum size of advice increases abruptly. In this case our bounds are asymptotically tight for topology recognition and asymptotically almost tight for spanning tree construction. We show that Θ(m log µ) bits of advice are enough and necessary to recognize topology in the class of graphs with m edges and multiplicity µ >. For the second task we show that Ω(µ log(n/µ)) bits of advice are necessary and O(µ log(m/µ)) bits of advice are enough to construct a spanning tree in the class of graphs with n nodes, m edges and multiplicity µ >. Thus in this case the gap between our bounds is always at most logarithmic, and the bounds are asymptotically tight, e.g., for multiplicity µ = O(n α ), where α is any constant smaller than. Our results imply the following, somewhat surprising comparison of the importance of graph exploration in accomplishing the two considered tasks. For the task of topology recognition, the fact that the agent can explore the graph has a small impact on the required size of advice, if µ >. Indeed, for any µ >, there are O(m log n) port-labeled non-isomorphic graphs with n nodes, m edges and multiplicity µ. Hence without any exploration it would be enough to give O(m log n) bits of advice to recognize topology (by giving the index of the graph in some ordered list of all such graphs), and with the help of exploration the number of bits is Θ(m log µ). Thus the capability to explore the graph is worth at most a logarithmic factor in the size of advice (the largest difference occurring when µ is constant). In other words, the fact that the agent can explore the graph decreases the size of advice needed for topology recognition only by a logarithmic factor, as compared to a motionless agent. By contrast, for the task of spanning tree construction, the possibility of exploring the graph may have a crucial impact on the required size of advice. Indeed, we prove that for any µ there are at least Ω(n/µ) graphs with n nodes and multiplicity µ, no pair of which has isomorphic spanning trees. Hence, without exploration, Ω(n/µ) bits of advice would be necessary to solve the spanning tree construction problem. However, if the agent can explore the graph, only O(µ log(m/µ)) bits are sufficient to find a spanning tree. Thus, for any µ polylogarithmic in n, the capability of exploring the graph is worth an exponential decrease of the size of advice required for spanning tree construction. In other words, an agent capable of exploring the graph requires exponentially smaller advice to solve the spanning tree construction problem, as compared with a motionless agent.. Related work Network algorithms with advice were studied, e.g., in [6, 7, 8, 9, 0,, 4, 0]. When advice is given by the oracle to the nodes, rather than to a mobile agent, the advice paradigm becomes closely related to that of informative labeling schemes [,,, 5, 6, 4]. In [] it was shown that giving appropriate -bit labels to nodes of a graph allows an agent with a constant-size memory to explore all graphs, and that with -bit labels an agent can explore all graphs of bounded degree. In the advice paradigm the authors studied the minimum size of advice required for the solvability of the respective network problem or for its efficient solution. In [8] the authors compared the minimum size of advice required to solve two information dissemination problems using a linear number of messages. In [9] the authors established the size of advice given to a mobile agent, needed to break competitive ratio of an

4 exploration algorithm in trees. In [0] it was shown that advice of constant size permits to carry on the distributed construction of a minimum spanning tree in logarithmic time. In [7] the authors established lower bounds on the size of advice needed to beat time Θ(log n) for -coloring of a cycle and to achieve time Θ(log n) for -coloring of unoriented trees. It was also shown that, both for trees and for cycles, advice of size Ω(n) is needed to -color in constant time. In the case of [0] the issue was not efficiency but feasibility: it was shown that Θ(n log n) is the minimum size of advice required to perform monotone connected graph clearing, using the minimum number of searchers. In [4] the authors studied radio networks for which it is possible to perform centralized broadcasting in constant time by a distributed algorithm. They proved that O(n) bits of advice allow to obtain constant time in such networks, while o(n) bits are not enough. In [] the trade-off between the size of advice and broadcasting time in trees was investigated, assuming that advice is given only to the source of broadcasting. In [6] the authors studied on-line computation with advice. Computability in anonymous networks and feasibility of distributed tasks performed using message exchange in anonymous networks, without advice, have been studied, e.g., in [, 8,, 5]. Rendezvous was considered in this context, e.g., in []. The concept of quotient graphs, which we use as defined in [5], has been studied extensively in different settings and under different names, such as minimum base. Boldi and Vigna provide a detailed description of fibrations, defined for the more general case of directed graphs [9]. In particular, they give a polynomial-time (off-line) algorithm that for a given graph computes its minimum base. Fibrations have been used e.g. in [7] for the leader election problem, which is a classical symmetry breaking problem, where it is assumed that the processors are indistinguishable and communicate over the network by exchanging messages. For some further work on the leader election problem and on the problem of assigning unique labels to the processors in a distributed system see [0,, 8]. The concept of quotient graph has been studied earlier in [] using the notion of coverings, for the scenario of message passing systems. Both coverings and fibrations are often used to determine the situations where finding a solution to some problem is impossible due to the symmetry of the graph. Informally speaking, if a base graph or a graph covered by the graph that models the network map several nodes of the network into the same class, then all those nodes perform the same computation, and therefore they cannot be distinguished. In our case nodes are indistinguishable for an agent exploring the network. Our proof techniques have some similarities to the ones used in the above-mentioned papers, as proving such results usually reduces to the construction of two graphs in which the execution of the algorithm is identical. However, in this work we deal with graph exploration by an agent, rather than with message passing systems and hence symmetry breaking issues are sometimes different (in the presence of a single agent the leader election problem becomes trivial). Métivier et al. [9] proved that some additional knowledge (the size of the graph) must be provided to the nodes of the graph to make several computational tasks feasible in the message passing model. Some of the techniques we apply, e.g. the idea of connecting independent isomorphic components with a cycle in order to obtain a graph of desired multiplicity (that we use in Lemma 4. and Proposition 5.), are similar to those present in [6, ]. Terminology and preliminaries Networks are modeled as simple undirected connected graphs (without self-loops or multiple edges). Nodes of a graph are unlabeled and ports at a node of degree d are arbitrarily labeled,..., d. Thus each edge has two labels, one at each extremity. We do not assume any coherence between port labelings at different nodes. Port labels are visible to an agent walking in the graph, that is, if the agent occupies a node v, then it can recognize the port labels at v for all edges incident to v, and whenever the agent arrives at a node, it knows both port labels of the edge it traversed. Graphs G = (V, E ) and G = (V, E ) are isomorphic, if there is a bijection f : V V, such that u is adjacent to v, if and only if f(u) is adjacent to f(v), and the port number corresponding to edge {u, v} at node u is equal to the port number corresponding to edge {f(u), f(v)} at node f(u), for distinct u, v V. An oracle is a function O from a class G of graphs to the set of finite binary strings. Given a graph G G, the oracle gives to the agent the string O(G), called the advice. The length of O(G) is called the size of advice. The agent (knowing O(G) but not G) is placed by an adversary in an arbitrary node of 4

5 the graph G and executes its deterministic algorithm, for which O(G) can be a parameter. We say that an agent solves the topology recognition problem (respectively the spanning tree construction problem) in a class G of graphs, with advice of size s, if for any graph G G, the agent, given advice of size s, outputs an isomorphic copy of G (respectively an isomorphic copy of a spanning tree of G) upon the execution of its algorithm. We will use the following notion from [5]. Let G be a graph and v a node of G. The view from v is the infinite rooted tree V(v) with labeled ports, defined recursively as follows. V(v) has the root x 0 corresponding to v. For every node v i, i =,..., k, adjacent to v in G, there is a child x i in V(v) such that the port number at v corresponding to edge {v, v i } is the same as the port number at x 0 corresponding to edge {x 0, x i }, and the port number at v i corresponding to edge {v, v i } is the same as the port number at x i corresponding to edge {x 0, x i }. Node x i, for i =,..., k, is now the root of the view from v i. We illustrate this definition in Fig., where Fig. (a) gives a graph G and Fig. (b) depicts the view from node c. (In this work we deal with node-unlabelled graphs, but we use labels in Fig. in order to make a clear correspondence between the nodes of G and the nodes of the view.) By V t (v) we (a) a b d c b d c a c b V(d) (b) b d c bb d d V(b) V(b) V(d) V(b) V(b) V(d) Figure : (a) a graph G; (b) the view V(c) denote the view V(v) truncated to depth t. The following proposition is proved in []. Proposition. ([]) For a n-node graph, V(u) = V(v), if and only if V n (u) = V n (v). The following notion was introduced in [5]. Let G be a graph. The quotient graph of G, denoted Q G, is a (not necessarily simple) graph defined as follows. Nodes of Q G correspond to the maximal sets of nodes of G which have the same view. For any (possibly equal) nodes x and y of Q G, corresponding to sets U and V, respectively, there is an edge between x and y with labels p at x and q at y, if and only if there exists an edge {u, v} in G with u U, v V and with ports p at u and q at v. Graphs G and H are called equivalent, if Q G = Q H. If G has n nodes and Q G has k nodes, then k divides n (cf. [5]) and the multiplicity µ of G is equal to n/k. Using the ideas from [5] it is possible to show that computing views of all nodes of G is the maximum information that can be obtained from exploring G by an agent. More precisely, we say that the executions of an algorithm by an agent starting in a graph G from node v and by an agent starting in a graph H from node w are identical, if the port labeled routes of both agents are the same and the degrees of the corresponding nodes in both routes are equal. We have the following proposition. Proposition. Let G and H be equivalent graphs with the quotient graph Q. Consider agents starting from node v of G and from node w of H, where v and w correspond to the same node of Q. If the agents have the same advice for G and H, then the executions of their algorithm are identical. Proof: Let A and A be two agents that execute the same algorithm and receive the same advice. Suppose that agents A and A start at nodes u 0 = v and u 0 = w of G and H, respectively, where u 0 and u 0 correspond to the same node in Q. Let P i and P i be the paths traversed by A and A, respectively, after i steps, where a step is traversing one edge of a graph. Let the ith node of P i (respectively P i ) be u i (respectively u i ) for each i 0. We prove by induction on the number of steps i performed by the agents that: (i) the port labels at u i in G are the same as the port labels at u i in H, and for the edges {u i, u i } 5

6 and {u i, u i } the port label at u i (respectively u i ) equals the port label at u i (respectively u i ) for each i > 0; (ii) u i and u i correspond to the same node of Q. This, together with the assumption that the advice given to the agents is the same, implies the proposition, because the input to the algorithms executed by A and A in step i is the advice and the corresponding paths P i and P i, respectively. The proof by induction goes as follows. If i = 0, then the claim follows from the assumption on u 0 and u 0. Suppose that the claim is true for i. The agents execute their algorithms to determine the next edge to follow. By assumption, the input to their algorithms is identical, so they select the edges {u i, u i+ } and {u i, u i+ } with the same port numbers at u i and u i. Moreover, u i and u i correspond to the same node of Q, which gives that the port numbers of the above mentioned edges are also the same at u i+ and at u i+. Since G and H are equivalent, u i+ and u i+ correspond to the same node of Q and therefore port labels at u i+ and at u i+ are identical in G and H, respectively. While topology recognition (and hence spanning tree construction) can be accomplished for any tree by an exploring agent working without advice, the following result shows that this is impossible for any graph that is not a tree. Theorem. For every algorithm A for an exploring agent that works without advice and every graph G that is not a tree, there exists a graph G, such that A can correctly accomplish neither the topology recognition nor the spanning tree construction task for both G and G. Proof: Let {u, v} be any edge of any cycle in G and let G {{u, v}} be the graph obtained from G by removing the edge {u, v}. Denote by G, G two copies of G {{u, v}} and let u i and v i be the nodes of G i corresponding to u and v of G, respectively, i =,. Construct G by taking G and G and by adding the edges {u, v }, {u, v } to G. For i =,, the port number at u i (resp. at v i ) corresponding to edge {u i, v i } is equal to the port number at u (resp. at v), corresponding to edge {u, v}. Note that G is connected. We prove that if w is a node in G and w is the corresponding node in G j, j {, }, then the view from w in G is identical to the view from w in G. To this end, we argue that there exists a path P i of length i in G starting at w, if and only if there exists an isomorphic path P i in G starting at w. (Recall the the isomorphism preserves port labels.) We proceed by induction on the length i of a path. Each path of length i = 0 consists of w in G and of w in G. Thus, let i > 0. Take any path P i starting at w in G and let P i be its initial segment of length i. By the induction hypothesis, there exists a path P i in G, starting at w, that is isomorphic to P i. Consider the last edge {x, y} of P i, where x is the endpoint of P i. Let x be the endpoint of P i. If {x, y} {u, v}, i.e. {x, y} is not the edge corresponding to the edges {u i, v i }, not present in G i, i =,, then there exists an edge {x, y } in G with the same port labels as {x, y} in G. On the other hand, if {x, y} = {u, v}, then assume w.l.o.g. that x = v and P i ends at v of the subgraph G of G, i.e. x = v (the other three cases when x {v, u, u } are analogous). Then, the port numbers of {u, v} and {x, u } are identical at u and x and at v and u. Thus, we extend P i with the edge {x, u } to obtain the desired path P i, isomorphic to P i. The reverse implication, where one extends P i to P i, provided that P i is given and P i is isomorphic to P i, can be proved in a similar way. We have shown that the quotient graphs of G and G have the same number of nodes and it remains to argue that they are isomorphic. Let {a, b} be an edge of the quotient graph of G. If {a, b} has been added to the quotient graph due to an edge e {u, v} of G, then the corresponding edge also belongs to the quotient graph of G, which follows immediately from the definition of G. If, on the other hand, e = {u, v}, then by construction the edge {u, v } belongs to G and has the desired port numbers at u and v. Thus, the corresponding nodes are also adjacent in the quotient graph of G, because, as argued above, the views of u and u as well as the views of v and v are identical. The fact that for each edge of the quotient graph of G there exists the corresponding edge in the quotient graph of G can be proved analogously. This shows that graphs G and G are equivalent. In view of Proposition., algorithm A, working without advice, has identical executions in G and G, when starting at corresponding nodes, e.g., respectively, at nodes u and u. Hence it cannot correctly accomplish neither the topology recognition nor the spanning tree construction task for both G and G because these graphs have different sizes. The next proposition shows that any upper bound on the size of the graph G is sufficient to construct Q G after exploring G (see also []). 6

7 Proposition. Given any upper bound n on the number of nodes of a graph G, there exists an algorithm for an exploring agent that finds the quotient graph Q G after exploring the graph G. Proof: Given n, the agent is able to construct the view V n (v), where v is the starting node of the agent. (Note that, in our model, the agent is able to retract its movements.) All views V n (u) for any node u of G are subgraphs of V n (v) and hence can be found by the agent. Since n > n, Proposition. implies that nodes with equal views can be correctly identified. It follows from the definition of the quotient graph that Q G can be now constructed. The next two propositions follow from the definition of the quotient graph. Proposition.4 ([5]) Let G be a graph with multiplicity µ. Let v i be the node in Q G corresponding to the set of nodes V i of G having the same view. Consider an edge e = {v i, v j } of Q G with corresponding port numbers p and q. If E e is the set of edges of G corresponding to e, then we have:. if i = j and p = q, then E e forms a perfect matching in V i,. if i = j and p q, then E e is a collection of pairwise disjoint cycles containing all the nodes in V i,. if i j, then E e is a perfect matching in V i V j such that no edge has both endpoints either in V i or in V j. Proposition.5 ([5]) Let G be any n-node graph of multiplicity µ and let T QG be any spanning tree of the quotient graph Q G. Then, there exist µ node-disjoint subtrees of G, each of which is isomorphic to T QG. Graphs with multiplicity In this section we show that for graphs of multiplicity, ω() bits of advice are enough to accomplish both the topology recognition and the spanning tree construction tasks (an arbitrarily slowly growing function of the size of the graph will do), but Θ() bits are not enough for either of these tasks. Lemma. There exists no algorithm for an exploring agent that can find the number of nodes of the graph in the class of graphs of multiplicity, provided that Θ() bits of advice are given. Proof: The idea of the proof is to construct a family of graphs, such that an exploring agent using advice of constant size behaves identically in two graphs of this family with a different number of nodes, and thus must output an incorrect number of nodes in one of them. We first construct recursively an infinite sequence of sets of graphs G i, i. The set G contains only the graph given in Fig. (a). Given G i, i, we construct G i+ as follows. Take a graph G G i and any integer j. Split the node s of G (see Fig. (b) for the result of splitting s in G ) to obtain a graph G, where the two nodes of G corresponding to s in G are denoted by s and t. Introduce j copies of G and connect them in a chain so that the node t of the k-th copy of G coincides with the node s of the (k + )st copy of G, k =,..., j. Finally, take a star on i + nodes and connect it with the chain in such a way that the central node of the star becomes adjacent to the node s of the first copy of G and to the node t of the last copy. The port numbers of the cycle have labels and (there is only one such labeling of the edges of the chain incident to the star that is consistent with the remaining labels of the chain), while the labels on the pendant edges of the star are selected arbitrarily. The resulting graph is denoted by ξ j (G). Let s in ξ j (G) be the node on the cycle at the largest distance from the unique star. Now we define G i+ = {ξ j (G): G G i, j N}. Let G = i G i. See Figs (c) and (d) for ξ (G) and ξ i (G), i, respectively, where G G. Each G G has multiplicity, because there is exactly one node of maximum degree in G. (The port labels at each node are,..., d, where d is the degree of the node, which implies that if G contains exactly one node v of maximum degree d, then the port label d is used once in G. Consequently, v has a unique view in 7

8 (a) 4 (b) s s t 4 (c) 4 s (d) 4 4 s Figure : The graphs (a) G G ; (b) G ; (c) ξ (G) G ; (d) ξ i (G) G, i G.) We say that a graph H is derived from the graph H, if either H = H or H = ξ i (ξ i ((ξ it (H)))), for some i,..., i t, where i j and t. Suppose that A is an algorithm for the agent that finds the number of nodes of a graph in G using advice of size Θ(). A receives q Θ() bits of advice and suppose that the initial position of the agent is the node s. The advice partitions G into at most q classes, G = C C q, such that the advice is the same for each graph in a given class. For each such partition there exists a set C k, k {,..., q }, such that for some H C k there exist infinitely many graphs ξ j (H), such that at least one graph derived from ξ j (H) is in C k. This holds, because otherwise we find a graph ξ j (H) such that none of the graphs derived from ξ j (H) belongs to the same class C k as H, and we apply the same argument for ξ j (H). The claim follows from the fact that the choice of ξ j (H) is always possible and the number of classes q is constant. Suppose that the explored graph is H. Assume that the agent operates in a steps when it starts at node s in H and then outputs the number of nodes of the explored graph. Let c be the length of the cycle in H. The agent traverses the cycle at most a/c times. By the assumption, a graph H derived from ξ b (H) is in C k, for some b > a/c. By the definition of H, the subgraphs induced by the nodes visited by the agent for H and for H cannot be distinguished by the agent. Thus, the agent outputs the same value when exploring H and H. If this value is correct for H, it must be incorrect for H. Corollary. There exists no algorithm for an exploring agent that recognizes topology or constructs a spanning tree in the class of graphs with multiplicity, provided that Θ() bits of advice are given. Proof: The existence of such an algorithm would contradict Lemma., because if an algorithm is able to determine the graph or its spanning tree, then the algorithm can output the number of nodes. If µ =, then G and Q G are isomorphic. Hence Q G contains all the information needed to recognize topology and to construct a spanning tree. Due to Proposition., it is sufficient to provide advice containing any upper bound on the number of nodes of G. Let ϕ: N N be any (computable) function diverging to infinity. (The divergence can be arbitrarily slow.) We construct an algorithm A ϕ that uses advice of size at most ϕ(n), where n is the number of nodes of the explored graph G, and determines the graph Q G. There exists a non-decreasing (computable) function ϕ : N N such that ϕ (n) ϕ(n) for all n, and ϕ diverges to infinity. Hence w.l.o.g. we may assume that ϕ itself is non-decreasing. For a n-node graph G the advice consists of the number x = ϕ(n) 8

9 and the information that µ =. Then A ϕ computes the smallest integer n, such that ϕ(n) > x. By the definition, n is an upper bound on n. By Proposition., using n, the algorithm A ϕ can find Q G. It outputs this graph for the topology recognition task. The correctness of A ϕ follows from the fact that Q G is isomorphic to G. Since the agent using the algorithm A ϕ recognizes the topology, it can also construct a spanning tree of G. Since the number ϕ(n) can be coded on less than ϕ(n) bits and one bit is enough to code µ =, we have: Theorem. For any (computable) function ϕ: N N such that ϕ = ω(), there exists an algorithm for an exploring agent that uses advice of size at most ϕ(n) and solves both the problem of topology recognition and that of the spanning tree construction in the class of n-node graphs of multiplicity. Together with Corollary., Theorem. gives the optimal size of advice to solve either of our two problems for graphs of multiplicity. 4 Topology recognition for graphs of multiplicity > In this section we establish asymptotically tight bounds on the size of advice needed for topology recognition in the class of graphs of any multiplicity µ >. Lemma 4. For any µ and m 4µ there exist µ Ω(m) equivalent non-isomorphic graphs of multiplicity µ, where m is a multiple of µ for µ > and is arbitrary for µ =. Proof: Fix the number of nodes n = rµ m/, where r. We define a graph B µ with n nodes. If µ >, then B µ is as in Fig. 4(a), while for µ = the graph B µ is given in Fig. 4(c). The quotient graph (a) (b) (c) (d) Figure 4: The graphs (a) B µ for µ > ; (b) Q Bµ for µ > ; (c) B ; (d) Q B corresponding to B µ is depicted in Fig. 4(b) in the first case and in Fig. 4(d) in the second case. The nodes of Q Bµ are denoted by v,..., v r. For each node v i, the corresponding set (of size µ) of nodes of B µ is V i. For each µ the multiplicity of B µ is µ. The number of edges in B µ is n for each µ > and n for µ =. In order to complete our construction, we will augment B µ with the remaining m n = Θ(m) edges, preserving the multiplicity of the graph. Different ways of augmenting B µ will lead to different resulting graphs. Augmenting the graph B µ will be done in steps. (Simultaneously, we augment the quotient graph Q Bµ.) The class of graphs resulting from the l-th step of the construction is denoted by ζ l (B µ ), and let ζ 0 (B µ ) = {B µ }. Step l consists in selecting two different nodes v i, v j of Q Bµ with minimum degrees and adding an edge {v i, v j } to Q Bµ. (The degrees of v i and v j increase by one in Q Bµ.) Set the two minimum (not yet used) port numbers on both sides of the edge. An operation of adding this edge to Q Bµ is reflected in graphs from ζ l (B µ ) by introducing a perfect matching between the nodes in V i and in V j, where each edge of this matching preserves the port labeling of {v i, v j } in Q Bµ. Moreover, each such new matching has to respect the existing edges in graphs from ζ l (B µ ), i.e., no multiple edges are allowed in graphs of the resulting class ζ l+ (B µ ) (as opposed to the situation in Q Bµ ). See Figure 5(a) for an example of the quotient graph of a graph B µ, where µ = 4, n = 6 and r = 4. The dotted line indicates an edge we use to augment Q Bµ. Figs 5(b)-5(f) depict some examples of augmentation of B µ. All those graphs are non-isomorphic and belong to ζ (B µ ). For µ = we also impose the restriction i j >, because otherwise there is only one possibility for placing the edges of 9

10 (a) (c) (e) (b) (d) (f) Figure 5: (a) an example of Q Bµ ; (b)-(f) different ways of augmenting B µ a perfect matching. Moreover, if for µ > and an integer k > 0 step k + would result in augmenting Q Bµ so that it has more than µ/ parallel edges between a pair of nodes, then we stop and ζ k (B µ ) is the final class of graphs. Also, we stop if m edges have been introduced to the initial graph B µ. (These restrictions will be used in counting graphs in ζ k (B µ ).) If the graphs in ζ k (B µ ) have less than m edges, then there are Θ(µ) edges between each pair of nodes in the quotient graph obtained in step k. There are Θ(r ) = Θ(n /µ ) such pairs. Thus, the number of edges in the quotient graph is Θ(n /µ). Each edge of the quotient graph corresponds to µ edges in a graph in ζ k (B µ ). Thus, regardless of the value of µ, a total of Θ(n ) edges can still be introduced in B µ in k steps respecting the above restrictions. Thus, the initial graph B µ can be augmented with Θ(m) edges in this way. Now we count the number of non-isomorphic graphs in the class ζ k (B µ ). The edge e added to Q Bµ in step l has different port numbers than the edges in B µ, because each edge of B µ has port number at one of its endpoints and no edge added during the augmentation of B µ can have port number, since this port number has been used at each node in the graph B µ. By Proposition.4 the multiplicity of any graph from ζ k (B µ ) equals µ and consequently, no isomorphism between graphs from ζ k (B µ ) can map a node from V i to a node from V j for i j. Suppose that an edge {v i, v j } has been added to Q Bµ in step l, l > 0. Let H ζ l (B µ ) be selected arbitrarily. For each node x in V i there exist Ω(µ) nodes in V j that are not adjacent to x in H. If the nodes of H were distinctly labeled, then there would be µ Ω(µ) possible matchings between them with the given restrictions. The nodes of H are not labeled, but the path containing only edges with port label on at least one of its endpoints, and connecting a node in V i with a node in V j in each graph in ζ l (B µ ) is contained in B µ. Thus, such a path between two nodes in V i and V j is unique with respect to the symmetry and to the O(µ) rotations of the unique cycle in B µ. Therefore, in each step there are µ Ω(µ) perfect matchings resulting in non-isomorphic graphs. Since each step adds exactly µ edges to B µ and, as argued above, B µ can be augmented with Θ(m) edges, we obtain a bound on the number of steps k = Θ(m/µ). Any two matchings added in the above way to non-isomorphic graphs from a class ζ l (B µ ) result in non-isomorphic graphs in the class ζ l+ (B µ ). This implies that there are µ Ω(m) non-isomorphic graphs in the class ζ k (B µ ). If the number of edges in each graph from ζ k (B µ ) is smaller than m, then we continue adding the edges (either loops or edges between different nodes v i, v j ) to Q Bµ and the corresponding edges (a union of disjoint cycles or a perfect matching, according to Proposition.4) to ζ k (B µ ), until these graphs have m edges. This extension is always possible. Indeed, if the degree of a node x in a graph H in ζ k (B µ ) is less than n, then there exists y V j in H such that {x, y} does not belong to H, j {,..., r}. Thus, each such node x has the corresponding unique node y in V j, because otherwise such nodes x would not have identical views. Therefore, if x / V j, then we can find the desired matching in H, while in the case when x V j there exists a desired union of cycles in the set V j. Denote the resulting class of graphs by ζ k (B µ). Note that when a cycle is introduced in this way, then the port labels on the cycle are different than those of the unique cycle in B µ, for µ >. Similarly, 0

11 the new non-loop edges added to Q Bµ can be distinguished from the edges added previously, by their port labels. This implies that for two non-isomorphic graphs in ζ k (B µ ) we obtain two non-isomorphic graphs in ζ k (B µ). Hence there are µ Ω(m) non-isomorphic graphs in the class ζ k (B µ). All of them have m edges and multiplicity µ. By construction, all graphs in ζ k (B µ) are equivalent. Finally note that if µ > then by Proposition.4 each edge of the quotient graph corresponds to exactly µ edges in G. On the other hand, if µ =, then an edge of the quotient graph may correspond to either a matching of size or to a single edge in G. Therefore, m is a multiple of µ for µ >, while m is arbitrary for µ =. Theorem 4. Every algorithm for an exploring agent that recognizes topology in the class of graphs with multiplicity µ and m 4µ edges requires Ω(m log µ) bits of advice. Proof: By Lemma 4. there exists a class G of equivalent and non-isomorphic graphs with multiplicity µ and with m edges such that G µ cm = cm log µ for some constant c > 0. Suppose that an agent is able to recognize the topology of G G using cm log µ bits of advice. This advice partitions the set G into k = cm log µ µ cm / disjoint classes C,..., C k with the same advice for all graphs in the same class. Thus, there exists i {,..., k} such that C i >. For different G, H C i, graphs G and H are equivalent and, since the advice is the same for G and H, the agent produces the same output for both graphs, in view of Proposition.. By Lemma 4., the graphs G and H are not isomorphic a contradiction. Together with Theorem 4., the following result gives an asymptotically tight bound on the size of advice sufficient for topology recognition in the class of graphs with m edges and multiplicity µ. Theorem 4. For any µ there exists an algorithm for an exploring agent that uses O(m log µ) bits of advice and recognizes topology in the class of graphs with m edges and multiplicity µ. Proof: Let n and r be the number of nodes of G and of its quotient graph Q G, respectively. By definition and by Proposition.4, n = rµ and Q G has Θ(m/µ) edges, where m is the number of edges of G. The oracle provides advice consisting of the multiplicity µ of G and of a sequence of m/µ sets S,..., S m/µ that the agent uses to obtain G. The set S i, i {,..., m/µ}, describes the edges of G corresponding to the i-th edge of Q G according to some fixed canonical order of the edges of Q G. This order is used both by the oracle and by the agent. Each edge {u, v} of G corresponding to the edge represented by S i is given as two integers c u and c v in {,..., µ}. Note that the size of this advice is O(m log µ), because S i µ for each i =,..., m/µ. The total number of integers c v given as the advice is an upper bound on the number of nodes of G. By Proposition., the agent is able to find Q G. The agent starts with an edgeless graph G whose nodes are partitioned into µ sets of size n/µ each, where the i-th such set will be called an i-th copy of Q G in G. For each i =,..., m/µ the agent uses S i to determine the set of edges of G corresponding to an edge {x, y} of Q G, where the port numbers at x and y are a and b, respectively. By Proposition.4, {x, y} corresponds to a matching M or to a union of cycles C in G. We consider the former case and the proof is identical for the set of cycles. (The only difference lies in the numbers of edges, but in both cases we obtain the same bound, because the result is asymptotic.) The agent needs to compute the pairs of the endpoints in M in the graph G, where one endpoint is in the set X of nodes (with the same view) corresponding to x while the other is in the set Y corresponding to y. For the j-th edge in the matching the agent uses the j-th element in S i to determine the copies of Q G connected by the edge of the matching. The intersections of X and Y with the corresponding node sets of the copies consist of one element. Therefore, the agents is able to determine the corresponding edge in G. Finally observe that the port numbers of this edge are a and b. The following observation shows that, for the problem of topology recognition, exploration is not worth much in terms of decreasing the size of advice. While Θ(m log µ) bits are needed with exploration, O(m log n) bits are enough without it, hence the ratio is at most logarithmic. This should be contrasted with the situation for spanning tree construction, where, as will be seen in Section 5, exploration is worth a lot.

12 Proposition 4. For any n and m there exists an algorithm for an agent that does not perform any exploration, uses O(m log n) bits of advice, and recognizes topology in the class of graphs with n nodes and m edges. Proof: The advice given to the agent consists of a list of m edges. Each edge {u, v} of G is given as four numbers in {,..., n}, i.e. u, v and the two port numbers at u and at v. 5 Spanning tree construction for graphs of multiplicity > In this section we establish asymptotically almost tight bounds Ω(µ log(n/µ)) and O(µ log(m/µ)) on the size of advice needed for the spanning tree construction in the class of n-node graphs of any multiplicity µ >. These bounds differ by at most a logarithmic factor and are tight for multiplicity µ = O(n α ), where α is any constant smaller than. Lemma 5. For every µ and for every n = rµ, where r is an integer, there exists a collection G of (n/µ) Ω(µ) n-node equivalent graphs with multiplicity µ, such that no two graphs in G have isomorphic spanning trees. Proof: First we analyze the case when µ < 8. Since the result is asymptotic, we can assume that n is sufficiently large, in particular n > 5µ. This implies that r > 5. Consider the quotient graph Q given in Fig. 6(b) (this example is for µ = and r = 5). (For even r take the quotient graph with r nodes and add a pendant edge arbitrarily.) The edge e with port numbers and 4 is unique in Q. The edge with port numbers and 4 at distance i from e is called the i-th edge of Q, i = 0,..., (r 5)/. Define G i by taking µ copies of Q and rearranging the µ edges corresponding to the i-th edge of Q into a perfect matching between their left and right endpoints that makes G i connected (see Fig. 6(a) for an example). Let G = {G 0,..., G (r 5)/ }. (a) (c) (b) v 4 Figure 6: (a) graph G G with µ = ; (b) quotient graph Q (c) the distinguished subgraph H of C Claim 5. If i j, then G i and G j do not have isomorphic spanning trees. Proof of the claim: A graph G k G contains as a subgraph a cycle C with edges in the perfect matching and edges with port labels and, connecting the endpoints of the edges in the matching. Moreover, exactly one edge of this cycle does not belong to a spanning tree of G k, for otherwise the tree is not connected. Thus, in particular, each spanning tree of G k contains a subgraph H of C depicted in Fig. 6(c). The node v of this subgraph uniquely identifies the length of the path in G k with endpoint v and containing the edges which do not belong to C and have port labels and. This determines the distance in the quotient graph between the edge corresponding to the perfect matching and the edge with port numbers, 4. This distance equals k, which means that if the spanning trees of G i and G j are isomorphic, then i = j, which proves the claim. Moreover, G = (r 5)/ = Θ(n). This, together with Claim 5., gives the bound from the lemma for each fixed µ, µ < 8. Let now 8 µ n. For simplicity of presentation, assume that µ is divisible by 4. At the end of the proof we show how to handle the general case. Let R be any family of (µ/4)-node paths rooted at an endpoint. Each edge e of R R has a label l(e) {,..., r}. Two rooted paths R and R are similar, if

13 there exists a bijection f from the set of nodes of R into the set of nodes of R such that {u, v} is an edge of R if and only if {f(u), f(v)} is an edge of R, and l({u, v}) = l({f(u), f(v)}) holds for all adjacent nodes u, v of R. (In other words, R and R are similar, if they have the same length and identical edge labelings up to symmetry.) For a rooted path R, we use notions of parent and child, as in any rooted tree. For a given rooted path R R we construct a graph G R of multiplicity µ. We will show that all graphs G R are equivalent and that for non-similar paths R the corresponding graphs do not have isomorphic spanning trees. The result will follow from the fact that there are many non-similar (µ/4)-node paths rooted at an endpoint. Before defining G R for R R, we introduce the basic building block, called the component, used to construct the graphs G R. Each component has 4r nodes. The component together with its quotient graph Q are given in Figs 7(a) and 7(b), respectively. The graph Q is also the quotient graph of each final (a) (b) z z 4 z zr zr (c) (d) (e) (f) z z Figure 7: (a) the component and (b) the quotient graph of the component and of G R for each R R for µ < n/; (c)-(d) all possible connections between different components; (e) the component and (f) the quotient graph for µ = n/ graph G R constructed below. We distinguish in the component an arbitrary maximal path on r nodes connected by edges with port labels in {, 4}. We will call this path the leading path of the component. The nodes of Q are denoted by z,..., z r (see Fig. 7(b)). Each node v of R is represented by a copy of the component in G R, denoted by G R (v). (See Fig. 8(a) and 8(c).) For any nodes u and v in R, such that u is the parent of v, we define c(u, v) to be if u has a parent w and l({w, u}) = l({u, v}), and to be 4 otherwise. If u is the parent of v in R, then we connect G R (u) with G R (v) in such a way that c(u, v) nodes in G R (u) corresponding to z l({u,v}) in Q form a cycle together with the unique node corresponding to node z l({u,v}) in Q that belongs to the leading path of G R (v). Fig. 7(c) depicts the four nodes of a component G R (v), corresponding to a node z i of Q, and connections representing the case when v has exactly one incident edge with label i and this edge connects v with its child, or its parent. Fig. 7(d) depicts these nodes and connections in the case when v has two incident edges with label i. By construction, the graph G R is connected. Moreover, G R has multiplicity µ, for each µ n/. For µ = n/, a similar construction can be used, except that it has to be ensured that each horizontal edge in the component has different labels at the endpoints (see Figs 7(e) and 7(f)). The rest of the proof is for µ n/, the case µ = n/ being analogous. See Fig. 8 for an example of the construction of G R. Fig. 8(a) depicts a path R with 4 nodes. We take µ = 6 and n = 48. The quotient graph Q given in Fig. 8(b) has r = nodes. The graph G R is shown in Fig. 8(c). We define G(R) = {G R : R R}. In order to prove the lower bound stated in the lemma, we count the number of graphs in G(R) that do not have isomorphic spanning trees. The proof is in two steps. First we show that if R is not similar to R, then G R and G R cannot have isomorphic spanning trees. This will follow from the path-like structure of the graphs in G(R). In this way, we reduce our task to estimating the number of non-similar paths that can form a family R. We begin with the following claim. Claim 5. If graphs G R and G R have isomorphic spanning trees, then R and R are similar.

GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

More information

The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

More information

Tree-representation of set families and applications to combinatorial decompositions

Tree-representation of set families and applications to combinatorial decompositions Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no

More information

Broadcasting in Wireless Networks

Broadcasting in Wireless Networks Université du Québec en Outaouais, Canada 1/46 Outline Intro Known Ad hoc GRN 1 Introduction 2 Networks with known topology 3 Ad hoc networks 4 Geometric radio networks 2/46 Outline Intro Known Ad hoc

More information

Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

More information

A 2-factor in which each cycle has long length in claw-free graphs

A 2-factor in which each cycle has long length in claw-free graphs A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

More information

Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

More information

6.852: Distributed Algorithms Fall, 2009. Class 2

6.852: Distributed Algorithms Fall, 2009. Class 2 .8: Distributed Algorithms Fall, 009 Class Today s plan Leader election in a synchronous ring: Lower bound for comparison-based algorithms. Basic computation in general synchronous networks: Leader election

More information

Offline sorting buffers on Line

Offline sorting buffers on Line Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

Graph Theory Problems and Solutions

Graph Theory Problems and Solutions raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

More information

Analysis of Algorithms, I

Analysis of Algorithms, I Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadth-first search (BFS) 4 Applications

More information

A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

More information

Outline 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs, part 1

Outline 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs, part 1 GRAPH THEORY LECTURE STRUCTURE AND REPRESENTATION PART A Abstract. Chapter focuses on the question of when two graphs are to be regarded as the same, on symmetries, and on subgraphs.. discusses the concept

More information

Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

More information

Full and Complete Binary Trees

Full and Complete Binary Trees Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

More information

Diversity Coloring for Distributed Data Storage in Networks 1

Diversity Coloring for Distributed Data Storage in Networks 1 Diversity Coloring for Distributed Data Storage in Networks 1 Anxiao (Andrew) Jiang and Jehoshua Bruck California Institute of Technology Pasadena, CA 9115, U.S.A. {jax, bruck}@paradise.caltech.edu Abstract

More information

most 4 Mirka Miller 1,2, Guillermo Pineda-Villavicencio 3, The University of Newcastle Callaghan, NSW 2308, Australia University of West Bohemia

most 4 Mirka Miller 1,2, Guillermo Pineda-Villavicencio 3, The University of Newcastle Callaghan, NSW 2308, Australia University of West Bohemia Complete catalogue of graphs of maimum degree 3 and defect at most 4 Mirka Miller 1,2, Guillermo Pineda-Villavicencio 3, 1 School of Electrical Engineering and Computer Science The University of Newcastle

More information

On the independence number of graphs with maximum degree 3

On the independence number of graphs with maximum degree 3 On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs

More information

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

More information

Fairness in Routing and Load Balancing

Fairness in Routing and Load Balancing Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria

More information

Lecture 1: Course overview, circuits, and formulas

Lecture 1: Course overview, circuits, and formulas Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik

More information

Efficient Recovery of Secrets

Efficient Recovery of Secrets Efficient Recovery of Secrets Marcel Fernandez Miguel Soriano, IEEE Senior Member Department of Telematics Engineering. Universitat Politècnica de Catalunya. C/ Jordi Girona 1 i 3. Campus Nord, Mod C3,

More information

CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 Solutions CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

More information

A Sublinear Bipartiteness Tester for Bounded Degree Graphs

A Sublinear Bipartiteness Tester for Bounded Degree Graphs A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartite

More information

Competitive Analysis of On line Randomized Call Control in Cellular Networks

Competitive Analysis of On line Randomized Call Control in Cellular Networks Competitive Analysis of On line Randomized Call Control in Cellular Networks Ioannis Caragiannis Christos Kaklamanis Evi Papaioannou Abstract In this paper we address an important communication issue arising

More information

Ph.D. Thesis. Judit Nagy-György. Supervisor: Péter Hajnal Associate Professor

Ph.D. Thesis. Judit Nagy-György. Supervisor: Péter Hajnal Associate Professor Online algorithms for combinatorial problems Ph.D. Thesis by Judit Nagy-György Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai

More information

Product irregularity strength of certain graphs

Product irregularity strength of certain graphs Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer

More information

Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

More information

The chromatic spectrum of mixed hypergraphs

The chromatic spectrum of mixed hypergraphs The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex

More information

136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 CHAPTER 4. INDUCTION, GRAPHS AND TREES 136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

The Graphical Method: An Example

The Graphical Method: An Example The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

More information

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

More information

CMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma

CMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma CMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma Please Note: The references at the end are given for extra reading if you are interested in exploring these ideas further. You are

More information

Midterm Practice Problems

Midterm Practice Problems 6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator

More information

On the k-path cover problem for cacti

On the k-path cover problem for cacti On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

More information

CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010

CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010 CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison Gomory-Hu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected

More information

INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

More information

Smooth functions statistics

Smooth functions statistics Smooth functions statistics V. I. rnold To describe the topological structure of a real smooth function one associates to it the graph, formed by the topological variety, whose points are the connected

More information

arxiv:1112.0829v1 [math.pr] 5 Dec 2011

arxiv:1112.0829v1 [math.pr] 5 Dec 2011 How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

More information

Degrees that are not degrees of categoricity

Degrees that are not degrees of categoricity Degrees that are not degrees of categoricity Bernard A. Anderson Department of Mathematics and Physical Sciences Gordon State College banderson@gordonstate.edu www.gordonstate.edu/faculty/banderson Barbara

More information

8. Matchings and Factors

8. Matchings and Factors 8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council,

More information

arxiv:cs/0605141v1 [cs.dc] 30 May 2006

arxiv:cs/0605141v1 [cs.dc] 30 May 2006 General Compact Labeling Schemes for Dynamic Trees Amos Korman arxiv:cs/0605141v1 [cs.dc] 30 May 2006 February 1, 2008 Abstract Let F be a function on pairs of vertices. An F- labeling scheme is composed

More information

The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

More information

Tiers, Preference Similarity, and the Limits on Stable Partners

Tiers, Preference Similarity, and the Limits on Stable Partners Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider

More information

System Interconnect Architectures. Goals and Analysis. Network Properties and Routing. Terminology - 2. Terminology - 1

System Interconnect Architectures. Goals and Analysis. Network Properties and Routing. Terminology - 2. Terminology - 1 System Interconnect Architectures CSCI 8150 Advanced Computer Architecture Hwang, Chapter 2 Program and Network Properties 2.4 System Interconnect Architectures Direct networks for static connections Indirect

More information

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms

More information

An example of a computable

An example of a computable An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

More information

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

More information

Network File Storage with Graceful Performance Degradation

Network File Storage with Graceful Performance Degradation Network File Storage with Graceful Performance Degradation ANXIAO (ANDREW) JIANG California Institute of Technology and JEHOSHUA BRUCK California Institute of Technology A file storage scheme is proposed

More information

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

More information

GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY

GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY CARL G. JOCKUSCH, JR. AND PAUL E. SCHUPP Abstract. Generic decidability has been extensively studied in group theory, and we now study it in

More information

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

More information

Approximation Algorithms

Approximation Algorithms Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

More information

Lecture 22: November 10

Lecture 22: November 10 CS271 Randomness & Computation Fall 2011 Lecture 22: November 10 Lecturer: Alistair Sinclair Based on scribe notes by Rafael Frongillo Disclaimer: These notes have not been subjected to the usual scrutiny

More information

Mathematical Induction

Mathematical Induction Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

More information

Lecture 5 - CPA security, Pseudorandom functions

Lecture 5 - CPA security, Pseudorandom functions Lecture 5 - CPA security, Pseudorandom functions Boaz Barak October 2, 2007 Reading Pages 82 93 and 221 225 of KL (sections 3.5, 3.6.1, 3.6.2 and 6.5). See also Goldreich (Vol I) for proof of PRF construction.

More information

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH 31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

3. Eulerian and Hamiltonian Graphs

3. Eulerian and Hamiltonian Graphs 3. Eulerian and Hamiltonian Graphs There are many games and puzzles which can be analysed by graph theoretic concepts. In fact, the two early discoveries which led to the existence of graphs arose from

More information

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

More information

Single-Link Failure Detection in All-Optical Networks Using Monitoring Cycles and Paths

Single-Link Failure Detection in All-Optical Networks Using Monitoring Cycles and Paths Single-Link Failure Detection in All-Optical Networks Using Monitoring Cycles and Paths Satyajeet S. Ahuja, Srinivasan Ramasubramanian, and Marwan Krunz Department of ECE, University of Arizona, Tucson,

More information

Disjoint Compatible Geometric Matchings

Disjoint Compatible Geometric Matchings Disjoint Compatible Geometric Matchings Mashhood Ishaque Diane L. Souvaine Csaba D. Tóth Abstract We prove that for every even set of n pairwise disjoint line segments in the plane in general position,

More information

Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

More information

How Asymmetry Helps Load Balancing

How Asymmetry Helps Load Balancing How Asymmetry Helps oad Balancing Berthold Vöcking nternational Computer Science nstitute Berkeley, CA 947041198 voecking@icsiberkeleyedu Abstract This paper deals with balls and bins processes related

More information

On end degrees and infinite cycles in locally finite graphs

On end degrees and infinite cycles in locally finite graphs On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel

More information

The LCA Problem Revisited

The LCA Problem Revisited The LA Problem Revisited Michael A. Bender Martín Farach-olton SUNY Stony Brook Rutgers University May 16, 2000 Abstract We present a very simple algorithm for the Least ommon Ancestor problem. We thus

More information

14.1 Rent-or-buy problem

14.1 Rent-or-buy problem CS787: Advanced Algorithms Lecture 14: Online algorithms We now shift focus to a different kind of algorithmic problem where we need to perform some optimization without knowing the input in advance. Algorithms

More information

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

More information

3. Mathematical Induction

3. Mathematical Induction 3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

More information

Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

More information

High degree graphs contain large-star factors

High degree graphs contain large-star factors High degree graphs contain large-star factors Dedicated to László Lovász, for his 60th birthday Noga Alon Nicholas Wormald Abstract We show that any finite simple graph with minimum degree d contains a

More information

1 Construction of CCA-secure encryption

1 Construction of CCA-secure encryption CSCI 5440: Cryptography Lecture 5 The Chinese University of Hong Kong 10 October 2012 1 Construction of -secure encryption We now show how the MAC can be applied to obtain a -secure encryption scheme.

More information

OPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION

OPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION OPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION Sérgio Pequito, Stephen Kruzick, Soummya Kar, José M. F. Moura, A. Pedro Aguiar Department of Electrical and Computer Engineering

More information

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

More information

Lecture 17 : Equivalence and Order Relations DRAFT

Lecture 17 : Equivalence and Order Relations DRAFT CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

More information

Definition 11.1. Given a graph G on n vertices, we define the following quantities:

Definition 11.1. Given a graph G on n vertices, we define the following quantities: Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define

More information

Key words. multi-objective optimization, approximate Pareto set, bi-objective shortest path

Key words. multi-objective optimization, approximate Pareto set, bi-objective shortest path SMALL APPROXIMATE PARETO SETS FOR BI OBJECTIVE SHORTEST PATHS AND OTHER PROBLEMS ILIAS DIAKONIKOLAS AND MIHALIS YANNAKAKIS Abstract. We investigate the problem of computing a minimum set of solutions that

More information

Metric Spaces. Chapter 1

Metric Spaces. Chapter 1 Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

More information

SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

More information

Reading 13 : Finite State Automata and Regular Expressions

Reading 13 : Finite State Automata and Regular Expressions CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model

More information

Faster Deterministic Gossiping in Directed Ad Hoc Radio Networks

Faster Deterministic Gossiping in Directed Ad Hoc Radio Networks Faster Deterministic Gossiping in Directed Ad Hoc Radio Networks Leszek Ga sieniec 1, Tomasz Radzik 2, and Qin Xin 1 1 Department of Computer Science, University of Liverpool, Liverpool L69 7ZF, UK {leszek,qinxin}@csc.liv.ac.uk

More information

Regular Expressions and Automata using Haskell

Regular Expressions and Automata using Haskell Regular Expressions and Automata using Haskell Simon Thompson Computing Laboratory University of Kent at Canterbury January 2000 Contents 1 Introduction 2 2 Regular Expressions 2 3 Matching regular expressions

More information

Lecture 2: Universality

Lecture 2: Universality CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal

More information

Distributed Computing over Communication Networks: Maximal Independent Set

Distributed Computing over Communication Networks: Maximal Independent Set Distributed Computing over Communication Networks: Maximal Independent Set What is a MIS? MIS An independent set (IS) of an undirected graph is a subset U of nodes such that no two nodes in U are adjacent.

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part

More information

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

More information

UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE

UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE ANDREW LUM ADVISOR: DAVID GUICHARD ABSTRACT. L(2,1)-labeling was first defined by Jerrold Griggs [Gr, 1992] as a way to use graphs

More information

5.1 Bipartite Matching

5.1 Bipartite Matching CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

More information

8.1 Min Degree Spanning Tree

8.1 Min Degree Spanning Tree CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

More information

Victor Shoup Avi Rubin. fshoup,rubing@bellcore.com. Abstract

Victor Shoup Avi Rubin. fshoup,rubing@bellcore.com. Abstract Session Key Distribution Using Smart Cards Victor Shoup Avi Rubin Bellcore, 445 South St., Morristown, NJ 07960 fshoup,rubing@bellcore.com Abstract In this paper, we investigate a method by which smart

More information

IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 3, JUNE 2011 709

IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 3, JUNE 2011 709 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 19, NO 3, JUNE 2011 709 Improved Bounds on the Throughput Efficiency of Greedy Maximal Scheduling in Wireless Networks Mathieu Leconte, Jian Ni, Member, IEEE, R

More information

Topology-based network security

Topology-based network security Topology-based network security Tiit Pikma Supervised by Vitaly Skachek Research Seminar in Cryptography University of Tartu, Spring 2013 1 Introduction In both wired and wireless networks, there is the

More information

Turing Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005

Turing Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005 Turing Degrees and Definability of the Jump Theodore A. Slaman University of California, Berkeley CJuly, 2005 Outline Lecture 1 Forcing in arithmetic Coding and decoding theorems Automorphisms of countable

More information

ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu

ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a

More information

Codes for Network Switches

Codes for Network Switches Codes for Network Switches Zhiying Wang, Omer Shaked, Yuval Cassuto, and Jehoshua Bruck Electrical Engineering Department, California Institute of Technology, Pasadena, CA 91125, USA Electrical Engineering

More information

Finite dimensional topological vector spaces

Finite dimensional topological vector spaces Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

More information

A o(n)-competitive Deterministic Algorithm for Online Matching on a Line

A o(n)-competitive Deterministic Algorithm for Online Matching on a Line A o(n)-competitive Deterministic Algorithm for Online Matching on a Line Antonios Antoniadis 2, Neal Barcelo 1, Michael Nugent 1, Kirk Pruhs 1, and Michele Scquizzato 3 1 Department of Computer Science,

More information