# Should Tables Be Sorted?

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Should Tables Be Sorted? ANDREW CHI-CHIH YAO Stanford Umversay, Stanford, Cahforma ABSTRACT Optmaahty questions are examined m the following information retrieval problem. Given a set S of n keys, store them so that queries of the form, "Is x E S?" can be answered quickly It is shown that m a rather general model including all the commonly used schemes, [lg(n + I)] probes to the table are needed m the worst case, prowded the key space is sufficiently large The effects of smaller key space and arbitrary encoding are also explored KEY WORDS AND PHRASES mformauon retrieval, lower bound, optmaahty, query, Ramsey's theorem, search strategy, sorted table CR CATEGORIES. 3 74, 4 34, 5 25, Introduction Given a set S of n distinct keys from a key space M = (1, 2,..., m}, a basic information retrieval problem is to store S so that membership queries of the form, "Is j in S?" can be answered quickly. Two commonly used schemes are the sorted table and the hash table. In the first case, a query can be answered in [lg(n + 1)1 probes by means of a binary search. ~ The hash table scheme has a good average-case cost, but requires O(n) probes in the worst case for typical hashing schemes. Looking at various alternative methods, one gets the feeling that ~log n probes must be necessary in the worst case, if the key space M is large and we only use about minimal storage space. Our purpose is to study the truth of this statement. The question is nontrivial, as the existence of hashing suggests the possibility of schemes drastically different from, and perhaps superior to, the sorted table. Before presenting technical results, let us try to put the subject of this paper in perspective. In the literature, efficient methods have been devised to perform various primitives in data manipulations [1, 8, 20]. For example, a sequence of n "DELETE," "INSERT," "MIN" instructions can be performed in O(n log n) time. In recent years, lower bounds to the complexity of these problems have also begun to receive attenuon (e.g., [7, 9, 15, 18]). Since effficient data structures may utilize the full power of a random-access machine (e.g., [20]), it is of special interest to study the complexity problems in general models that are equipped with some address-computing capabilities. This paper is one step in that direction, studying perhaps the simplest of such data structuring problems. It Is hoped that one can derive interesting results for other problems in similar frameworks. (For related study regarding bitwise-random-access machines, see [5, 6, 10].) lg denotes logarithm with base 2 Permission to copy without fee all or part of this material is granted provided that the copies are not made or dlstnbuted for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery To copy otherwise, or to republish, reqmres a fee and/or specific permission This research was supported in part by the National Science Foundation under Grant MCS Author's address: Department of Computer Science, Stanford University, Stanford, CA ACM ll/81/ \$00.75 Journal of the Association for Computing Machinery, Vol 28, No 3, July 1981, pp

2 616 ANDREW CHI-CHIH YAO sorted table 11,2t, {2,3} ~ ~ {2,31 {1,3} = {1,31 2 probes cyclic table " L.!_12_1 - L I2_I -l_2_u 1 probe FIG 1 The sorted table is not optzmal for n = 2, m = 3 2. The Wisdom of Using Sorted Tables In this section we show that for large key space, [lg(n + 1)] probes are required to answer the membership problem in a rather general model. This model encompasses all common schemes such as hashing, sorted tables, and linked list structures. For clarity, we first prove the result m a simplified model. The general result will be given in Theorem 1'. The Basic Model. Let the key space be M = (1, 2... m }. We are interested in storing n distinct keys of M into a table of size n. A table structure J-specifies how any particular set of n keys are to be placed in the table T. A search strategy 5~ corresponding to J-specifies, for any given key K, how to perform a series of probes TOO =?, T(iz) =?... into the table T, until one can claim whether K is in T or not. The search strategy ls fully adaptive, in the sense that each probing location can depend on K and on all the previous probing results. The cost c(j, 5 e) of a (table structure, search strategy) pair is measured by the number of probes needed in the worst case. The complexityf(n, m) is the minimum cost achievable by any such pair. Clearlyf(n, m) _< [lg(n + 1)]. To get some feehng for possible improvements over the sorted table scheme and on its ultimate limitation, we look at the simple case n = 2, m = 3. It is easy to see that two probes are needed to decide whether K = 2 is in T if a sorted table is used. However, the "cyclic" table in Figure 1 allows us to answer any query in just one probe, as the first entry of T determines the entire table. Note that these are the only two nomsomorphlc table structures (up to the renaming of keys and table locations) for this case. Thus, a sorted table is not optimal for n = 2, m = 3. We now show, however, that a sorted table is optimal as soon as n = 2, m -- 4 (hence for all n -- 2, m > 4). Any table structure for n = 2, m = 4 can be uniquely represented as a directed graph on four labeled vertices { 1, 2, 3, 4}. We draw an edge i ~ j if the pair {i, j} is stored as [i-~. For example, the graph in Figure 2 represents a table structure with { 1, 4) stored as ~ and (2, 4) as ~--~. For any three vertices in the graph, the edges between them may or may not form a directed cycle. It is not hard to show that for any such graph on four vertices, there exist three vertices among which the edges are acyclic. In Figure 2, { 1, 3, 4} is such a set of three vertices. If we consider the set of keys corresponding to these verhces as a subspace with m = 3, we find that we are storing these keys as a "permuted" sorted table, that is, one that differs from the sorted table only in a new ordering 3 < 1 < 4 of the elements (Figure 3). But this means that any searching strategy for this table structure must make two probes m the worst case. This proves that f(2, 4) _> 2, and hence the sorted table is optimal for n = 2, m >_ 4.

3 Should Tables Be Sorted? /1",,, 3 131,1 FIG 2 A typical table FIG 3 The "permuted'" structure for n = 2, m = 4 sorted table corresponding to {1, 3, 4) from Figure 2 The preceding statement generalizes to any fixed n; that is, the sorted table scheme Is optimal for any fixed n, provided that the key space Is large enough. THEOREM 1. For every n there exists an N(n) such that f(n, m) = [lg(n + l)]for all rn >_ N(n). PROOF. argument. We need the following lemma, which can be proved by an adversary LEMMA 1. If a table structure stores the keys of a table m sorted order (or according to some fixed permutatwn), then [lg(n + 1)] probes are needed in the worst case by any search strategy provided that rn >_ 2n - 1 and n >_ 2. PROOF. We will construct an adversary strategy to show that [lg(n + 1)] probes are required to search for the key value K = n of the space {1, 2... m}. The construction ~s by induction on n. For n -- 2 and m _ 3 it is easy to see that two probes are required. Let no > 2. Assume the induction hypothesis to be true for all n < no; we will prove it for n = no, m - 2no - 1, and K = no. By symmetry, assume that the first probe position p satisfies p _< [no/2]. The adversary answers T(p) = p. Then the key no may be in any position i, where [no/2] + 1 i no. In fact, T([no/2] + 1) through T(no) is a sorted table of size n' = [no/2j which may contain any subset of {[no/2] + 1, [no/2] m}, and hence, in particular, any subset of the key space M' = {[no/2] + 1, [no/2] m - [no/2]}. The size m' of M' satisfies and the desired key no has relative value K' = no - [n0/2] = n' in the key space M'. By the induction hypothesis, [lg(n' + 1)] more probes will be reqmred. Hence the total number of probes is at least 1 + [lg(n' + 1)] = 1 + [lg([n0/2j + 1)] _> [lg(n0 + 1)]. This completes the induction step. [] To prove Theorem 1, the idea is to show that If m IS large enough, then for any table structure Jthere is a set So of 2n - I keys with the following property: Given any n-key subset A C So, the table structure always arranges the keys of A according to some fixed permutation. Lemma 1 will then imply the [lg(n + 1)] bound. To this end, let us partition d, the family of n-key subsets of M, into n! parts as follows. For each A = {J1 < j2 <.-- < jn} E d, let Ta be the table formed under ~-. We assign A to the group o(6, tz... in) if Ta(il) = J1, Ta(iz) = J2... TaOn) = jn. The collection {o(6, t2... in)[(ll, t2... in) is a permutation of {1, 2... n}} forms a partmon of d.

4 618 ANDREW CHI-CHIH YAO CLAIM. If m is sufficiently large, then there exists a set of 2n - 1 keys So _C (1, 2... m } such that for all n-key subsets A _C So we have A ~ o(il, i2,.., in), where (il, i2... in) is a fixed permutation. By our earlier discussion this would imply Theorem 1. It remains to prove the claim. We make use of the following famous combinatorial theorem (see, e.g., [3]). RAMSEY'S THEOREM. For any k, r, t, there exists a finite number R(k, r, t) such that the following is true. Let S = (1, 2... m) with m _> R(k, r, t). If we divide the family of all r-element subsets of S into t parts, then at least one part contains all the r- element subsets of some k elements of S. Our claim follows from Ramsey's Theorem by choosing r = n, t = n! and k = 2n - 1. This proves Theorem 1 with N(n) = R(2n - 1, n, n!). [] Generalization. As we mentioned at the beginning of this section, Theorem 1 holds under more general conditions. In the general setting a table may contain "pointers" and duplicated keys. Formally, we have a universe M of m keys, a set P ofp special symbols (pointers), and an array T containing q cells. Let S..C M be any subset of n keys. We store S in T, where each cell may contain any element in the set S O P. Each key in S may appear several times, or may not appear at all. A rule for determining the above assignment is a table structure ~. Defining search strategies ~as before, we measure the cost c (J, ~) by the number of probes needed to answer the membership query in the worst case. The complexityf(n, m, p, q) is the minimum cost achievable by such a pair. THEOREM 1'. For any n, p, q, there exists an N(n, p, q) such that f(n, m, p, q) = [lg(n + 1)] for all m _> N(n, p, q). PROOF. As the proof is very similar to that of Theorem 1, we shall only sketch it. Clearly we need only prove thatf(n, m, p, q) _> [lg(n + 1)] for all large m. Let 3-be any table structure. To each n-key subset S we assign a q-tuple (il, i iq) with 1 _< it _< n + p, where iz = k if T[I] contains the kth smallest key in S and h = n + j if T[I] contains thejth pointer. This partitions the family of all n-key subsets into (n + p)q classes. If m _> R(2n - 1, n, (n + p)q), then by Ramsey's theorem there exists a set So of 2n - 1 keys all of whose n-key subsets are in the same class. By definition, all tables for n-key subsets S C So contain identical pointers in each location, and hence tables are distinguished only by the keys stored in the tables. Now, in these tables the set of locations containing a given key depends only on the relative ranking of the key in the n-key subset. Therefore, from the viewpoint of search strategies, these are sorted tables (with possible missing keys). By Lemma 1, it takes [lg(n + 1)] probes in the worst case. As ~-is arbitrary, this proves the theorem. [] We may further allow the set S to have nonunique representations as a table (as is the case of hash tables and search trees), since this obviously will not ~mprove the worst case cost. Thus the present model allows for the use of linked lists, search trees, all common hashing techniques, etc. 3. When Is One Probe Sufficient? The numbers N(n) in Theorem 1 are extremely large even for moderate n. Thus the result is not too useful in practical terms. It is of interest to understandf(n, m) for smaller m. We therefore ask the following equivalent question: Given n, k, what is the maximum m such thatf(n, m) = k? Call this number g(n, k). Hence, if and only

5 Should Tables Be Sorted? 619 tenants room FIG The assooatlon between tenants and rooms m the proof of Theorem 2 if there are more than g(n, k) possible keys, then we have to use more than k probes in the worst case. The determination ofg(n, k) ts difficult, but we can determine it in one special case. THEOREM 2 = [ 3 /f n = 2, g(n, 1) 2n-2 /f n>2. L PROOF. We shall give a proof for the lower bound to g(n, 1) by exhibiting a l- probe table structure for the asserted number of keys. The other part of the proof, that is, that no table structure can achieve a l-probe search for a larger key space, involves a lengthy case analysis and is left to Appendix A. For the case n = 2, m = 3, the "cyclic" table discussed earlier has an obvious l- probe search strategy. Now let n > 2 and m = 2n - 2; we describe a table structure allowing a l-probe search strategy. Consider the situation as m people sharing an apartment building with n rooms. We need a method so that no matter which n people appear at the same time, we can assign them in such a way that it is possible to determine if person j is present by looking up the occupant of one particular room (dependent on j). We use K~ to stand for the personj (1 _<j _~ m). Let us call Kj and K,+~ the tenants of room j, for 1 _< j <_ n - 2; Kj is the lower tenant and Kn+j the upper tenant. For room n - 1, K,~-i is a lower tenant, and for room n, Kn is a lower tenant. There are no upper tenants for these two special rooms. (See Figure 4.) When a group of n people shows up, we make the assignment by the following steps. (i) If room j (1 ~ j _< n - 2) has only one tenant present, assign that tenant to the room. (ii) Ifa roomj (1 <_j _< n - 2) has both tenants present, let the upper tenant go to a room which has no tenants present. (iii) Those people left unassigned are either tenants whose upper tenants are also present, or are keys K,-1, K,. We assign them so that they do not occupy the rooms of which they are tenants (e.g., a cyclic shift will do). The last step can always be accomphshed, for we can argue that if there is at least one person left in (iii), then there are at least two. Indeed, either (a) assuming neither Kn-~ nor K, is present, then at least two rooms j (1 _<j _< n - 2) have both tenants present; (b) assuming exactly one of K,-1, K, is present, then there must be another j (1 ~_j _< n - 2) with both tenants present; or (c) both K,-1 and Kn are present. For example, assume in Figure 5 that the group (1, 2, 3, 6, 7, 9, 10, 12) shows up. Steps (i)-(iii) are illustrated. To answer if Kj is in the table, we look at the room of which it is a tenant. (a) If Kj is there, then it is m the table. (b) If an upper tenant of some other room is there, then Kj is not in the table. (c) If a lower tenant of some other room is there, then Kj is in the table.

6 620 ANDREW CHI-CHIH YAO step (i) step (,,) step (m) FIG 5 An illustration of steps (O-On) in the assignment FIG 6 Fadure of the i-probe scheme with 2n - 1 keys It ts straightforward to verify the correctness of the answers. This proves g(n, 1) _> 2n - 2 for n > 2. It remains to prove the upper bounds for g(n, 1). We have shown that g(2, 1) < 4 in Section 2. The proof ofg(n, 1) <_ 2n - 2 for n _> 3 is left to Appendix A. [] Note that the above scheme only allows one to determine in one probe whether an element is in the table, but not the location of the element (when it is in). It would be interesting to study schemes that can determine the stored locations of the elements and to determine how large a key space can be accommodated under this stronger requirement. Remark. It Is somewhat surprising that the l-probe schemes used in the above proof are optimal, as they look quite arbitrary. In particular, why do we need two special rooms n - 1 and n? Figure 6 shows that the scheme fails if we have only one special room (and 2n - 1 keys). The arrival of keys 1, 2... n - 1, n + 1 will make the accommodation impossible. 4. Searching in Two Probes How strong is Theorem 1'? It appears to be a robust result, considering its generality. However, the following surprising result demonstrates that it depends heavily on the fact that keys outside of the set S may not be present m the table. THEOREM 3. There exists a number N'(n) such that if m _> N'(n), then by adding one extra cell in a sorted table, the search can always be accomplished tn two probes. (The content in the extra cell is allowed to be any integer between l and m.)

7 Should Tables Be Sorted? f sorted table.a. N 621 cell n FIG 7 A 2-probe table PROOF. We define a concept called "k-separating systems." Let M = (1, 2... m) and n > 0 an integer. An n-separator F = (A1, A2... An) is an ordered n-tuple of subsets A, _C M which are mutually disjoint. An n-separating system for M is a famdy of n-separators such that for any n elements xl < x2 < < x, of M, there exists (not necessarily unique) a member F = (A1, A2... An) E ~with x, E As for i = l, 2... n. Let us use ~b(xl, x2... xn) to denote this F. For y E U]=i Aj, use J(F, y) to denote the j with y E A~. We now show how to design a 2-probe structure with the help of an n-separating system W for M. Let W= {F1, F2... Fl}. For each n-tuple a = (xl < Xz <... < Xn) drawn from M, let F, ta) = ~b(xl, xz... Xn). For the moment assume that ] ~[ = l _< m. We organize the table as shown in Figure 7. To test lfa numbery E M is in the table, one first probes at cell 0 to find i(a), then makes a second probe at position J(F,(a), y). The numbery is in the table if and only if it is in this location. Reason. Let F~(a) = (A1, A2... A,); if y is in the table, then y E Aj with j = J(F,(~), y), and hence must be in thejth cell. It remains to examine the condition that l < m. We need the following combinatorial lemma. LEMMA 2. There exists an n-separating system ~for S with [~[ _< 4 n2. (Ig m) n-1. PROOF. See Appendix B. [] It follows from the lemma that if4n2.(lg m) 0-~ _< m, then the 2-probe scheme works. The condmon is satisfied if m _> N'(n) = 216"2. This proves Theorem 3. [] Bob Tarjan [private communication] has improved the bound N'(n) in Theorem 3 to exp(cn log n) by a somewhat different construction. In the proof of Theorem 3, the table structure used has a "directory" at cell 0. To retrieve a key y, one consults the directory to probe a cell which would contain j if and only ify is in the table. (Tarjan's construction also follows this pattern.) It is of interest to find tight bounds on m, n for such table structures (call them canomcal 2- probe structures) to exist. Define a primmve n-separating system ~ for M = (1, 2... m} to be a family of n-separators such that for any n distinct elements Xl, x2... x, of M, there exists a member F = (At, A2... A~) ~ ~with each As containing exactly one xj. Let b(m, n) be the minimum size of such a primitive n- separating system. It can be shown that m _> b(m, n) is a necessary and sufficient condition for a canonical 2-probe structure to exist. Ron Graham [private communication] has shown that asymptotically b(m, n) _< ~n enlog m by a nonconstructwe argument which implies the existence of a canonical 2-probe structure whenever m _> exp(cn) for some constant c > Conclustons We have discussed the complexity of the "membership" retrieval problem. The main conclusions are, roughly, that when the word size is large, sorted tables are optimal

8 622 ANDREW CHI-CHIH YAO structures if only the addressing power of a random-access machine can be used, but far from optimal once arbitrary encoding of the information is allowed in the table. These results are mainly of theoretical interest, although Theorem 3 suggests that there may be fast retrieval schemes in more practical situations. The Ramsey type technique used in the proof of Theorem 1 may have wider applications. Ron Rivest [private communication] has used it to prove a conjecture concerning [13]. Below we mention some subjects for future research. We have proved the optimality of sorted tables in a rather general framework (Theorem 1'). It would be nice if the threshold value N(n) could be substantially lowered. Also, the exact determination of quantities such as g(n, 2) poses challenging mathematical questions. When arbitrary encoding was allowed, we obtained a rather curious result (Theorem 3). In either of the extreme cases m ~ n and m _> 216n2, one needs at most two probes to decide if an item is in a table. In the former case the addressing power, and in the latter case the encoding power, contribute to fast retrieval. It would be interesting to study the problem for intermediate values of m. Tarjan and Yao [19] have shown that when m grows at most polynomially in n, one can retrieve in O(1) probes with a O(n)-cell table. The question is still open for other ranges of m, say, m = 2 "~. Another direction of research is to study the effect of restricting the decoding procedures. A main theme of this paper has been to discuss the membership problem in a word-length-independent framework (by letting m ~ oo). We list some open problems of prime importance in this framework, which are indirectly related to the membership problem. (1) It is easy to construct similar models for more complex data manipulation problems such as executing a sequence of "INSERT," "DELETE," "MIN." We conjecture that, unlike the membership problem, nonconstant lower bounds exist even if arbitrary encoding is allowed. (2) The Post Office Problem [4, 14]. Consider n points vl, v2... vn on an m m lattice (with m --> oo). Can we encode them m cn cells so that, given any point on the lattice, we can find the nearest v, in O(1) probes? In fact, this problem is unresolved even in the one-dimensional case. (3) Sorting Networks. In the usual Boolean networks for sorting n inputs in {0, 1), it is known [1 l] that one need only use O(n) gates A, V, 7. If we consider gates that are functions from M M to M, can we build a sorting network for n inputs from M with O (n) gates as m --~ oo? In general, the study of such networks for function computation would be interesting. See Vilfan [21] for some discussions on the formula size problems. Appendix A. Proof of Optimality in Theorem 2 In this appendix we complete the proof of Theorem 2 by showing that g(n, 1) _< 2n - 2 for n _> 3. For convenience, the inductive proof is organized in the following way. We first prove that for any n _> 3 and m = 2n - 1, a table structure allowing a 1-probe search induces a 1-probe table structure for n' = n - 1 and m' -- 2n' - 1. Then we demonstrate that for n = 3 and m = 2n - 1 = 5, there cannot be any l- probe table structure. This immediately implies g(n, 1) < 2n - 1 for all n _> 3, completing the proof. Suppose there is a l-probe table structure J- for n, m -- 2n - 1, where n _> 3. For 1 _<j _< 2n - 1, let lj be the location to examine when keyj is to be retrieved. Clearly, some location will be lj for at least two distinct j. Without loss of generality, assume

9 Should Tables Be Sorted? 623 that 11 =/2 = 1, that is, the content in T(I) determines whether key 1 and/or key 2 are in the table. For i = 1, 2 let Y, denote the set of keysj such that T(I) =j implies the presence of key i in the table, and let N, = (1, 2... m} - Y,. Certainly, T(1) ~ N~ if and only if key i is not in the table. Note that 1 ~ Yx and 2 E Y2. We distinguish four possibilities: Case I. 2 E Y1, 1 E Y2; CaseII. 2~Ni, IEN2; Case III. 2 ~ Y1, 1 ~ N2; Case IV. 2 ~ N1, 1 E Y2. We shall show that these cases either are impossible or imply the existence of a 1- probe table structure for n' = n - l and m' = 2n' - I. The following simple fact is relevant. FACT A1. [N~l>_n- lfori= 1,2. PROOF. Otherwise, let Y~ C Y, - (i) with I Y,'I = n. The table T storing Y,' will have T[1] E Y,, contradicting the absence of key i. [] LEMMA A1. Case I is impossible. PROOF. By Fact AI, INiI -> n - 1. Let xa, x2... xn-~ E Na. Then the set (1, x~, xz... Xn-a} cannot be satisfactorily arranged in a table T. A key xj in cell 1 would imply the absence of key 1, and key 1 in cell 1 would imply the presence of key 2. [] LEMMA A2. Case H is impossible. PROOF. By Fact A1, I Nal -> n - 1. Let 2, Xl, xz... xn-2 ~ Na. Then the set {1, 2, xl, x2... xn-2) cannot be arranged in a table T. A key xj or 2 in cell 1 would imply the absence of key 1, and key 1 in cell 1 would imply the absence of key 2. [] LEMMA A3. Cases III and IV both imply the existence of a 1-probe table structure for n' = n - 1 and m' = 2n' - 1. PROOF. We need only prove the lemma for case III; case IV merely switches the roles of keys 1 and 2 in case III. CLAIM A1. IN21 = n- 1. PROOF. By Fact AI, IN~I >- n - 1. Suppose IN~I > n - 1; let 1, x,, x2... xn-1 ~ N2. Then there is no way to accommodate {2, Xl, x2... xn-,} in a table T. A key xj in cell 1 would imply the absence of key 2, and key 2 in cell 1 would imply the presence of key 1. We conclude that I N2I = n - 1. [] Because of Claim AI we can write N2 = {1, 3, 4... n} and Y2 = (2, n + 1, n n - 1}, renaming the keys in {3, n - 1} if necessary. CLAIM A2. Y~ = (1, 2). PROOF. Otherwise, let ( 1, 2, x} _C Y~. If x E (3, 4... n}, then we cannot arrange the set (x, n + l, n n - l} in T, since T[l] = x would imply the presence of key 1 and T[I] = n +j would imply the presence of key 2. Ifx ~ (n + 1, n + 2,... 2n - 1), then we cannot arrange the set (x, 2, 3... n} in T by a similar reasoning. [] It follows from Claim A2 that Na = (3, n - 1}.

10 624 ANDREW CHI-CHIH YAO test for keys 1,2 test for keys 1,2 tl,3,4t T: tl,4,5~ -- tl,3,5f FIG 8 A parual configurauon for the l- probe table structure FIG 9 Our knowledge about the table structure after taking Claim A5 mto consideration CLAIM A3. In a table T formed from an n-key subset { 1, Xl, X2... Xn-1}, where xj ~ 2for all j, key 1 always appears in cell 1. PROOF. Otherwise T[1] = x~ for some j, implying the absence of key 1. [] CLAIMA4. For 3 _< j_< 2n -- 1, lj # 1. PROOF. By Claim A3, any n-key subset So with 1 E So, 2 ~ So will have key 1 in cell 1. Therefore, the key stored in T[ 1 ] cannot decide ifj E So. [] Consider the set of tables for storing all the n-key subsets { 1, xl, x2... XR-1} wlth Xj # 2 for allj. Because of Claims A3 and A4, cell 1 always contains key 1, and if we eliminate cell 1 from all these tables, we are left with a 1-probe table structure for all the (n - l)-key subsets of {3, n - 1}. This proves Lemma A3. [] We have completed the first part of the proof for g(n, 1) _< 2n - 2, namely, the existence of a l-probe table structure for n, m = 2n - 1 (n _> 3) implies the existence of such a structure for n' = n - 1, m' = 2n' - 1. It remains to prove that no 1-probe table structure exists for n = 3, m = 5. Assume that such a structure exists; we proceed to demonstrate a contradiction. By the preceding analysis we can assume that 11 = lz --- 1,/3, /4, /5 ~ 1, Y1 = { 1, 2}, N1 = {3, 4, 5}, Y~ = {2, 4, 5}, and N2 = { 1, 3}. As the naming of keys 4 and 5 is still arbitrary, we can assume that the tables storing sets {1, 3, 4}, {1, 3, 5), {1, 4, 5} are as shown in Figure 8. (Note that key 1 has to be in cell 1, and the remaining have to be in a cyclic order.) Next consider how the table structure arranges S = {2, 3, 4} and {2, 3, 5}. Keys 2 and 3 cannot be in cell 1 because T[I] = 2 would imply 1 E S and T[I] = 3 would imply 2 ~ S. Thus the arrangements can only be and {2, 3, 4} ~ either (a) (4, 2, 3), or (b) (4, 3, 2), {2, 3, 5} ~ either (a)' (5, 2, 3), or (b)' (5, 3, 2), where (z, j, k) means that cells 1, 2, 3 contain keys i, j, k, respectively. There are four possibilities, namely, (a) (a)', (a) x (b)', (b) (a)', and (b) (b)'. CLAIM A5. Only (b) x (a)' may be possible.

11 Should Tables Be Sorted? 625 test for test for test for keys 1,2 key5 key 4 test for test for test for keys 1,2 key5 key FIG 10 More knowledge about FIG 11 Adding (2, 5, 1) and the table structure (2, 1, 4) to the structure PROOF. If (a) X (a)' or (b) x (b)', then one cannot test in one probe whether key 4 is in the table (recall that/4 # l). If (a) (b)', again one cannot test in one probe whether key 4 is in the table -- if/4 = 2, then the tables (1, 3, 4) and (5, 3, 2) cannot be distinguished, and if/4 = 3, then (l, 5, 3) and (4, 2, 3) cannot be distinguished. Therefore, the table structure must contain the tables shown in Figure 9. [] How is the (3, 4, 5} arranged as a table? One cannot put key 4 or 5 into cell l since that would imply the presence of key 2. Also, the arrangement as (3, 4, 5) would make it impossible to test for key 3 (since there is a (l, 4, 5)). Thus it has to be arranged as (3, 5, 4). We now assert that/.5 = 2 and/4 = 3. Testing for key 5 at cell 3 cannot distinguish (l, 3, 4) and (3, 5, 4), and testing for key 4 at cell 2 cannot distinguish (l, 5, 3) and (3, 5, 4). Our knowledge about the l-probe table structure thus far is summarized in Figure 10. To fill in the slots for (1, 2, 4} and (l, 2, 5}, we note that key 2 has to be put into cell l since both keys l and 2 are here. The only posslbdity for (1, 2, 5} is (2, 5, 1); the alternative (2, l, 5) would jeopardize the test for key 4, since (1, 4, 5) is already there. This also means that T[3] = l implies the absence of key 4. It follows that (l, 2, 4} has to be arranged as (2, l, 4). The known part of the table structure is shown in Figure I I. However, there is now no way to test for key 3! If we probe at cell 2, the two tables (3, 5, 4) and (2, 5, l) cannot be distinguished; if we probe at cell 3, the tables (l, 3, 4) and (2, l, 4) wdl look the same. This contradicts the defimtlon of a table structure allowing a l-probe search strategy. We have thus proved that no l-probe table structure can exist for n = 3, m = 5. This completes the proof for g(n, l) < 2n - l (n _> 3) and hence Theorem 2. [] Appendix B. Proof of Lemma 2 Let m _> k _> 2 and S -- (1, 2... m}. We shall construct a k-separating system Y for S such that [~1-4k2(lgm) k-l. We agree that the O-separating system is, and the l-separating system for any T is {T}. The system,~ will be recurslvely constructed m the lexicographic order of (k, m). Divide S consecutively into k almost equal blocks Sa, \$2... Sk with IS, ] = m, -- [(m + i - l)/kl. We define ~ as the umon of the following famdles of k-

12 626 ANDREW CHI-CHIH YAO separators, to be described in a moment: d and ~(nl, n2... nd, where 0 _< n, < k are integers satisfying ~, n, = k. Let F, -- (A,x, A,2..., A,D be a k-separator for the set S, (1 _< i _< k). The direct sum Fi ~ Fz Fk is the k-separator (Aa, A2... AD, where Aj -- I.,I,A~j. Let t > 0 and, for each 1 _< i_< k, ~ -- {F,~, F,2... F,t} be a family of k-separators for S,. Define the dtrect sum ~2 ~k to be the family of k-separators for S,,.~ -- (F1, F2... Ft}, where Fj = F2j for 1 _< j _< t. We now construct alas follows. Let ~ (1 _< t _< k) be a k-separating system for S,, constructed recursively. 2 For each j, add arbitrary k-separators into ~ so that the resulting family ~; has t = max, [~ I elements. We now define d = ~g. For each xl < x2 < < xk there is clearly a k-separator F -- (A1, A2... AD ~ d that "separates" the x's (i.e., such that xj ~ Aj for all j) if all xj are in the same block S,. For each (nl, n2... nk) that satisfies 0 _< n, < k and ~,n, -- k, the family of separators ~(nl, n2... nk) is constructed as follows. The family ~(m, n2... nd is empty if there is some i such that n, > m,. Otherwise, for each 1 _< i _< k let ~" be an n,-separating system for S,, recursively constructed. Denote by ~(nx, n2,..., nd the family of all k-separators of the form F = (A~, A~2... Ai~, A2~... A2,~... Akn~), where each (A~, A,2... A,n,) ~ ~". For any x~ < x2 < < Xk in S such that exactly n, of the x's are in S, for each i, there is clearly some k-separator in ~(m, n2..., nd that separates the x's. Let ~ = ~/O (Un,~(nl, n2... nd). Then ~ is a k-separating system for S, as implied by the properties of ~ and ~ stated above. Let f~(n) denote the size of constructed this way. Then by definition, fi(m) = max,fi + ~ H fn,(m,), (l:])} O~n~<k t~l ~n,=k for m_>k>_2. (B1) We adopted in (B1) the convention that fo(mo = 1 andfn,(m,) = 0 if n, > m,. FACT B1. For each k _> 2, f~(m) is a nondecreasingfunction of re. PROOF. Using (BI), one can prove it by induction on (k, m), lexicographically. [] We shall now prove, by induction on k, the following formula: 3 f~(m) _< 4kZ(lg m) k-~ for m _> k _> 1. (B2) The formula is obviously true for k = 1. Let k > l; we shall prove (B2) assuming that it is true for all smaller values of k. First we prove the following fact. FACT B2. For m = k t, where t _> l is an integer, we have fk(m) --< 4k~(½ lg m) ~-~. PROOF. Using (B1), Fact Bl, and the induction hypothesis, we have fk(m)<--fk(~) +~o_<n, <k fo~ ~a, 4X'n~(lg m)x'~n'-~)" (B3) ~n,=k In (B3), the summations ~' are over those i with n, # O. The second term in (B3) is at most (2kk_-ll) 4t~-~)2+l(lg m)k-2 < ~ 1.4~2_k+2(lg m)k_2. 2 We agree that.~, = ~5 if k > I S, I. Also note that when k > I S, l, any k-separator (A~, A2,, Mk) for S, must have some A~ = ~5 a We mterpret 0 to be 1

13 ShouM Tables Be Sorted? 627 Thus (B3) implies _<~ +~.2.4 k~ lgm l < - (lo~m) 4 k~ lg m -2 m) k-1 _< 4k2 (~ lg [] For general m, let k t-i _< m < k t where t >_ 2. By Facts B1 and B2, fk(m) ~. 4 k2 lg k t < 4k~(lg m) k-1. This completes the inductive proof for (B2) and hence Lemma 2. [] A Bibliographic Note. The complexity of the membership problem was first raised in Minsky and Papert [10, pp ], where it was called the exact match problem. The model was formulated on a bitwise-access machine, with the complexity defined as the average number of bits needed to be examined for a random table. This model, especially the n = 1 case, was further examined by Elias and Flower [61, but the problem has not been solved completely even for this special case. Wordwise-access models were used in several recent papers. Sprrugnoli's work [16] dealt with efficient hash functions and is closely related to the materials in Section 4 of the present paper. Taljan [171 showed that tables of size O(n) and retrieval time O(log*n) can be achieved if m is at most polynomial in n; the retrieval time was improved to O(1) by Ta~an and Yao [19. Also see Bentley et. al [2[, Gonnet [7], and Munro and Suwanda [121 for other recent studies on related problems. ACKNOWLEDGMENT. I wish to thank Bob Tarjan for many helpful comments, which led me to include Theorem l' in the paper. REFERENCES 1 AHO, A V., HOPCROFT, J.E, AND ULLMAN, J D The Destgn and Analysis of Computer Algornhms Addison-Wesley, Reading, Mass., BENTLEY, J, DETIG, D, GUIBAS, L., AND SAXE, J. An opttmal data structure for mmnnal-storage dynamic member searching Unpublished manuscript. 3 BERGE, C Graphs and Hypergraphs North-Holland, Amsterdam, DOBKIN, D., AND LIPTON, R J Multidimensional search problems SIAM J Comput 5 (1976), ELIAS, P Efficient storage and retrieval by content and address of static files J. A CM 21, 2 (April 1974), ELIAS, P., AND FLOWER, R A The complexity of some smaple retrieval problems J A CM 22, 3 (July 1975), GONNET, G H Average lower bounds for open-addressing hash coding. Proc Conf on Theoretical Computer Science, Waterloo, Ontario, Canada, August 1977, pp KNUTH, DE TheArtofComputerProgrammmg, Vol 1. FundamentalAlgomhms Addision-Wesley, Reading, Mass, 1968

14 628 ANDREW CHI-CHIH YAO 9 KNUTH, D E The Art of Computer Programming, Vol 3 Sorting and Searching. Addlslon-Wesley, Reading, Mass, 1973 l0 MINSKY, i., AND PAPERT, S Perceptrons. MIT Press, Cambridge, Mass, MULLER, D E, AND PREPARATA, F P Bounds to complexities of networks for sorting and for switchmg J ACM 22, 2 (Aprd 1975), MUNRO, J I, AND SUWANDA, H Imphot data structures Proc ilth Ann. ACM Symp on Theory of Computing, Atlanta, Georgia, 1979, pp RIVEST, R.L Optimal arrangement of keys in a hash table J. ACM 25, 2 (April 1978), SHAMOS, M l. Geometric complexity Proc 7th Ann ACM Symp. on Theory of Computmg, Albuquerque, N M, 1975, pp SNYDER, L On uniquely representable data structures Proc 18th Ann IEEE Symp on Foundations of Computer Science, Providence, R I, 1977, pp SPRUGNOLI, R Perfect hashing functions A single probe retrieving method for static sets Commun ACM 20, 11 (Nov 1977), TARJAN, R E Storing a sparse table Stanford Computer Science Dep Rep STAN-CS , December 1978 (this is a preliminary version of [19]) 18 TARJAN, R.E A class of algorithms which require nonlinear time to maintain disjoint sets J Comput Syst Sct. 18 (1979), TARJAN, R E, AND YAO, A C Storing a sparse table Commun ACM 22, 11 (Nov 1979), VAN EMDE BOAS, P, KAAS, R, AND ZIJLSTRA, E Design and implementation of an efficient priority queue Math Syst Theory 10 (1977), VILFAN, B Lower bounds for the size of expressions for certain functions In d-ary logic Theor Comput Sct 2 (1976), RECEIVED JULY 1979, REVISED JUNE 1980, ACCEPTED AUGUST 1980 Journal of the Assooatlon for Computing Machinery, Vol 28, No 3, July 1981

### Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### On an anti-ramsey type result

On an anti-ramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider anti-ramsey type results. For a given coloring of the k-element subsets of an n-element set X, where two k-element

### 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.,

### CS 4310 HOMEWORK SET 1

CS 4310 HOMEWORK SET 1 PAUL MILLER Section 2.1 Exercises Exercise 2.1-1. Using Figure 2.2 as a model, illustrate the operation of INSERTION-SORT on the array A = 31, 41, 59, 26, 41, 58. Solution: Assume

Answers to Homework 1 p. 13 1.2 3 What is the smallest value of n such that an algorithm whose running time is 100n 2 runs faster than an algorithm whose running time is 2 n on the same machine? Find the

### CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992

CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons

### Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range

THEORY OF COMPUTING, Volume 1 (2005), pp. 37 46 http://theoryofcomputing.org Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range Andris Ambainis

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004

Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February

### A Note on Maximum Independent Sets in Rectangle Intersection Graphs

A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,

### 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

### CS 840 Fall 2016 Self Organizing Linear Search

CS 840 Fall 2016 Self Organizing Linear Search We will consider searching under the comparison model Binary tree lg n upper and lower bounds This also holds in average case Updates also in O(lg n) Linear

### 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

### Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### On the Union of Arithmetic Progressions

On the Union of Arithmetic Progressions Shoni Gilboa Rom Pinchasi August, 04 Abstract We show that for any integer n and real ɛ > 0, the union of n arithmetic progressions with pairwise distinct differences,

### 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)

### Learning Threshold Functions with Small Weights using Membership Queries

Learning Threshold Functions with Small Weights using Membership Queries Elias Abboud Research Center Ibillin Elias College P.O. Box 102 Ibillin 30012 Nader Agha Research Center Ibillin Elias College P.O.

### COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics

### Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay

Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding

### Brendan D. McKay Computer Science Dept., Vanderbilt University, Nashville, Tennessee 37235

139. Proc. 3rd Car. Conf. Comb. & Comp. pp. 139-143 SPANNING TREES IN RANDOM REGULAR GRAPHS Brendan D. McKay Computer Science Dept., Vanderbilt University, Nashville, Tennessee 37235 Let n~ < n2

### 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

### 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

### ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

### SECTION 10-2 Mathematical Induction

73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

### Regular Languages and Finite Automata

Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a

### Section 6-2 Mathematical Induction

6- Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### 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

### Universal hashing. In other words, the probability of a collision for two different keys x and y given a hash function randomly chosen from H is 1/m.

Universal hashing No matter how we choose our hash function, it is always possible to devise a set of keys that will hash to the same slot, making the hash scheme perform poorly. To circumvent this, we

### 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

### Design and Analysis of Algorithms

Design and Analysis of Algorithms CSE 5311 Lecture 3 Divide-and-Conquer Junzhou Huang, Ph.D. Department of Computer Science and Engineering CSE5311 Design and Analysis of Algorithms 1 Reviewing: Θ-notation

### 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

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### Storing a Sparse Table with O(1) Worst Case Access Time

Storing a Sparse Table with O(1) Worst Case Access Time MICHAEL L. FREDMAN AND J/~NOS KOMLOS University of Califorma, San Dtego, La Jolla, Cahforma AND ENDRE SZEMERI3DI Untverstty of South Carohna Abstract.

### 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

### 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

### Comparing Leaf and Root Insertion

30 Research Article SACJ, No. 44., December 2009 Comparing Leaf Root Insertion Jaco Geldenhuys, Brink van der Merwe Computer Science Division, Department of Mathematical Sciences, Stellenbosch University,

### AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC

AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC W. T. TUTTE. Introduction. In a recent series of papers [l-4] on graphs and matroids I used definitions equivalent to the following.

### Chapters 1 and 5.2 of GT

Subject 2 Spring 2014 Growth of Functions and Recurrence Relations Handouts 3 and 4 contain more examples;the Formulae Collection handout might also help. Chapters 1 and 5.2 of GT Disclaimer: These abbreviated

### Notes on Complexity Theory Last updated: August, 2011. Lecture 1

Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look

### 3. Recurrence Recursive Definitions. To construct a recursively defined function:

3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### NON-CANONICAL EXTENSIONS OF ERDŐS-GINZBURG-ZIV THEOREM 1

NON-CANONICAL EXTENSIONS OF ERDŐS-GINZBURG-ZIV THEOREM 1 R. Thangadurai Stat-Math Division, Indian Statistical Institute, 203, B. T. Road, Kolkata 700035, INDIA thanga v@isical.ac.in Received: 11/28/01,

### Two classes of ternary codes and their weight distributions

Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight

### 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:

### Linear Codes. Chapter 3. 3.1 Basics

Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

### Catalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni- versity.

7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni- Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,

### Odd induced subgraphs in graphs of maximum degree three

Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A long-standing

### 8.7 Mathematical Induction

8.7. MATHEMATICAL INDUCTION 8-135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### The Relative Worst Order Ratio for On-Line Algorithms

The Relative Worst Order Ratio for On-Line Algorithms Joan Boyar 1 and Lene M. Favrholdt 2 1 Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark, joan@imada.sdu.dk

### The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge,

The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge, cheapest first, we had to determine whether its two endpoints

### Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### More Mathematical Induction. October 27, 2016

More Mathematical Induction October 7, 016 In these slides... Review of ordinary induction. Remark about exponential and polynomial growth. Example a second proof that P(A) = A. Strong induction. Least

### Lecture 2: Analysis of Algorithms (CS )

Lecture 2: Analysis of Algorithms (CS583-002) Amarda Shehu September 03 & 10, 2014 1 Outline of Today s Class 2 Big-Oh, Big-Omega, Theta Techniques for Finding Asymptotic Relationships Some more Fun with

### Sample Induction Proofs

Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given

### New Hash Function Construction for Textual and Geometric Data Retrieval

Latest Trends on Computers, Vol., pp.483-489, ISBN 978-96-474-3-4, ISSN 79-45, CSCC conference, Corfu, Greece, New Hash Function Construction for Textual and Geometric Data Retrieval Václav Skala, Jan

### PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

### Notes on counting finite sets

Notes on counting finite sets Murray Eisenberg February 26, 2009 Contents 0 Introduction 2 1 What is a finite set? 2 2 Counting unions and cartesian products 4 2.1 Sum rules......................................

### Two General Methods to Reduce Delay and Change of Enumeration Algorithms

ISSN 1346-5597 NII Technical Report Two General Methods to Reduce Delay and Change of Enumeration Algorithms Takeaki Uno NII-2003-004E Apr.2003 Two General Methods to Reduce Delay and Change of Enumeration

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### [2], [3], which realize a time bound of O(n. e(c + 1)).

SIAM J. COMPUT. Vol. 4, No. 1, March 1975 FINDING ALL THE ELEMENTARY CIRCUITS OF A DIRECTED GRAPH* DONALD B. JOHNSON Abstract. An algorithm is presented which finds all the elementary circuits-of a directed

### 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].

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### TAKE-AWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION

TAKE-AWAY GAMES ALLEN J. SCHWENK California Institute of Technology, Pasadena, California L INTRODUCTION Several games of Tf take-away?f have become popular. The purpose of this paper is to determine the

### Determination of the normalization level of database schemas through equivalence classes of attributes

Computer Science Journal of Moldova, vol.17, no.2(50), 2009 Determination of the normalization level of database schemas through equivalence classes of attributes Cotelea Vitalie Abstract In this paper,

### Mathematical Induction. Lecture 10-11

Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach

### i/io as compared to 4/10 for players 2 and 3. A "correct" way to play Certainly many interesting games are not solvable by this definition.

496 MATHEMATICS: D. GALE PROC. N. A. S. A THEORY OF N-PERSON GAMES WITH PERFECT INFORMA TION* By DAVID GALE BROWN UNIVBRSITY Communicated by J. von Neumann, April 17, 1953 1. Introduction.-The theory of

### Problem Set #1 Solutions

CS161: Design and Analysis of Algorithms Summer 2004 Problem Set #1 Solutions General Notes Regrade Policy: If you believe an error has been made in the grading of your problem set, you may resubmit it

### 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

### Mathematical Induction

Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural

### Chapter 7. Induction and Recursion. Part 1. Mathematical Induction

Chapter 7. Induction and Recursion Part 1. Mathematical Induction The principle of mathematical induction is this: to establish an infinite sequence of propositions P 1, P 2, P 3,..., P n,... (or, simply

### 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

### OPTIMAL SELECTION BASED ON RELATIVE RANK* (the "Secretary Problem")

OPTIMAL SELECTION BASED ON RELATIVE RANK* (the "Secretary Problem") BY Y. S. CHOW, S. MORIGUTI, H. ROBBINS AND S. M. SAMUELS ABSTRACT n rankable persons appear sequentially in random order. At the ith

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### 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

### For the purposes of this course, the natural numbers are the positive integers. We denote by N, the set of natural numbers.

Lecture 1: Induction and the Natural numbers Math 1a is a somewhat unusual course. It is a proof-based treatment of Calculus, for all of you who have already demonstrated a strong grounding in Calculus

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

### Is Sometime Ever Better Than Alway?

Is Sometime Ever Better Than Alway? DAVID GRIES Cornell University The "intermittent assertion" method for proving programs correct is explained and compared with the conventional method. Simple conventional

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### Module 6: Basic Counting

Module 6: Basic Counting Theme 1: Basic Counting Principle We start with two basic counting principles, namely, the sum rule and the multiplication rule. The Sum Rule: If there are n 1 different objects

### Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

### On the Relationship between Classes P and NP

Journal of Computer Science 8 (7): 1036-1040, 2012 ISSN 1549-3636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,

### Lecture 2 February 12, 2003

6.897: Advanced Data Structures Spring 003 Prof. Erik Demaine Lecture February, 003 Scribe: Jeff Lindy Overview In the last lecture we considered the successor problem for a bounded universe of size u.

### Introduction to Algorithms. Part 3: P, NP Hard Problems

Introduction to Algorithms Part 3: P, NP Hard Problems 1) Polynomial Time: P and NP 2) NP-Completeness 3) Dealing with Hard Problems 4) Lower Bounds 5) Books c Wayne Goddard, Clemson University, 2004 Chapter

### Planar Tree Transformation: Results and Counterexample

Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem

### Collinear Points in Permutations

Collinear Points in Permutations Joshua N. Cooper Courant Institute of Mathematics New York University, New York, NY József Solymosi Department of Mathematics University of British Columbia, Vancouver,

### 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

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

### Optimal Binary Space Partitions in the Plane

Optimal Binary Space Partitions in the Plane Mark de Berg and Amirali Khosravi TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, the Netherlands Abstract. An optimal bsp for a set S of disjoint line segments

### Analysis of Algorithms I: Binary Search Trees

Analysis of Algorithms I: Binary Search Trees Xi Chen Columbia University Hash table: A data structure that maintains a subset of keys from a universe set U = {0, 1,..., p 1} and supports all three dictionary

### THE SETTLING TIME REDUCIBILITY ORDERING AND 0 2 SETS

THE SETTLING TIME REDUCIBILITY ORDERING AND 0 2 SETS BARBARA F. CSIMA Abstract. The Settling Time reducibility ordering gives an ordering on computably enumerable sets based on their enumerations. The