UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis

Transcription

1 UCSD CSE 1, Spring 01 [Section B00] Mathematics for Algorithm and System Analysis Lecture 17 Class URL:

2 Lecture 17 Notes Last time Recurrences revisited Types of graphs: Complete, Tree, Bipartite, Today and Next Tuesday Shortest path tree, minimum spanning tree, search trees Chromatic number Paths, Trails, Circuits, Cycles: Eulerian and Hamiltonian graphs ( complexity zoo NP, P, NP-hard, NP-complete, ) (most of the slides are in this slide deck already please read ahead!) Logistics Final exam review Sunday June 8, 6-8pm, PCYNH 106 Practice final exam questions early Week 10 Week 9 and Week 10 quizzes off-line Today s quiz: open on WeBWorK from 9:0am 10pm Can get up to 5 points out of Extra credit for PPT slide(s) (up to % of course grade) See next slide Things to Know end of Week 9, end of Week 10 SPIS slides on Trees, Trees, Trees, Please treat as assigned reading

3 Extra Credit Opportunity: PPT slide(s) on graph(s) 1% course grade for each slide, up to a max of See thread on Piazza Must use the given PPT template, and include: Image Source / attribution (URL) Brief description (what do vertices / edges represent, is the graph directed / undirected, etc.) Brief explanation of why the image and the graph are interesting Deadline = Thursday, June 5, 5pm Pacific time Turn-in: .pptx/.ppt attachment to natalie.lynn.larson@gmail.com Filename: LastName_FirstName_LastTwoDigitsofPID[_].pptx

4 Special kinds of graphs complete / clique Complete (simple) graphs: all possible edges

5 Special kinds of graphs bipartite Bipartite graphs Two kinds of vertices V 1 V = V, V 1 V = Each edge is between a vertex of one kind and a vertex of the other kind: any e E has form {u V 1, v V } People Projects

6 Trees Special kinds of graphs tree Connected V - 1 edges No cycles (any two of these properties implies the third!) Assignment: read the SPIS-01 Trees, Trees, Trees slides that are posted!

7 Trees vs. General Graphs Graph: G = (V,E) V = set of vertices E = set of edges graph with 7 vertices, 6 edges This graph is not connected, and it has a cycle. Connected: Between any pair of vertices, there is a path of edges. Cycle: A path of edges that begins and ends at the same vertex. 7

8 Tree = Special Kind of Graph A tree is a graph with n vertices that: (i) is connected (ii) has no cycles (iii) has n 1 edges tree with 7 vertices, 6 edges Fact: Any two of the above properties implies the third. E.g., If a graph with n vertices is connected and has no cycles, then it has n 1 edges. Can we prove this???

9 You want proof? I ll give you proof!

10 Claim: If a graph with n vertices connected and has no cycles, then it has n 1 edges. Proof Strategy:? Strong Induction (I.H.) Assume that the Claim is true for n = 1,, k 1. Consider a graph with n = k vertices that is connected and has no cycles. (k 1 1) + (k 1) + 1 = (k 1 + k ) 1 = k 1 k 1 k 1 vertices, connected, no cycles k 1 1 edges k k 1 vertices, connected, no cycles k 1 edges

11 Binary tree: Trees = Data Structures Has a root vertex Every vertex has at most children Binary Search Tree Application: Search a set of keys for a given key value if key = root.key then success else if key < root.key then search left subtree else search right subtree Search for Search for 17 Search for 8

12 How Many Search Steps Are Needed? How many levels can a binary tree with n nodes have? (convention: root is at level 0) At most:? At least:? n 1 levels log n levels

13 Trees = Data Structures = Algorithms A heap is a binary tree of depth d such that (1) all nodes not at depth d-1 or d are internal nodes each level is filled before the next level is started () at depth d-1 the internal nodes are to the left of the leaves and have degree, except perhaps for the rightmost, which has a left child each level is filled left to right A max heap (min heap) is a heap with node labels from an ordered set, such that the label at any internal node is ( ) the label of any of its children All root-leaf paths are monotone

14 Minheap Example Data Structure Parent-child arithmetic = how we d store in an array! Parent parent of A[i] has index = i div Left, Right (children) children of A[i] have indices i, i+1 Parent Child this is a minheap

15 Heap Operation: Insert Insert(S,x): O(height) O(log n) Keep swapping with parent until heap condition satisfied

16 Heap Operation: ExtractMin ExtractMin(S): return head, replace head key with the last key, float down O(log n) Float down : If heap condition violated, swap with smaller child

17 Insert + ExtractMin =? Priority queue abstract data type Always have the highest-priority item ready at hand Useful in algorithm implementation e.g., Dijkstra s algorithm later today Unordered list: Insert = O(1) but ExtractMin = O(n) Ordered list: ExtractMin = O(1) but Insert = O(n) Heap: Insert = O(log n), ExtractMin = O(log n)

18 O(n log n) Heapsort Build heap: n x O(log n) time for i = 1..n do insert (A[1..i],A[i]) Extract elements in sorted order: n x O(log n) time for i = n.. do [extract-min] Swap (A[1] A[i]) Heapsize = Heapsize-1 Float down A[1]

19 Trees = Solutions to Real Problems (A) Connecting cities with minimum road construction (when adding new junctions is allowed) (B) Calculating minimum-cost routes to destinations

20 The Minimum Spanning Tree Problem Given: graph of cities and distances B B A C A C E D E D Problem: Make all cities reachable from each other with minimum road construction.

21 The Shortest Paths Problem B B A C A C E D E D Problem: Compute the shortest routes from A to every other city.

22 The Shortest Paths Problem E A D B C Problem: Compute the shortest routes from B to every other city. E A D B C 5 0 What do these green numbers represent?

23 The Shortest Paths Problem E A D B C Exploration parties set out from B, always moving at constant speed. When a party is the first to reach some city, they call home, then split up and continue exploring. The times of the phone calls are the shortest-path distances from B. E A D B C 5 0 (You learn this as Dijkstra s Algorithm)

24 Minimum Spanning Tree, Again E A D B C E A D B C Start from an arbitrary vertex, and add the shortest incident edge to the growing tree that does not complete a cycle. (If the MST is unique, then the starting vertex doesn t matter ) (You learn this as Prim s Algorithm)

25 Nearly the Same Algorithm! Prim: Iteratively add the edge e(u,v) to T, where u is in T and v is not yet in T, to minimize d u,v (d u,v = (u,v) distance) Dijkstra: Iteratively add the edge e(u,v) to T, where u is in T and v is not yet in T, to minimize d u,v + l u (l u is u s shortest path cost) Both algorithms build trees, in similar greedy ways! Prim s Algorithm constructs a Minimum Spanning Tree Dijkstra s Algorithm constructs a Shortest Path Tree If Prim s Algorithm and Dijkstra s Algorithm had a baby Iteratively add the edge e(u,v) to T, where u is in T and v is not yet in T, to minimize d u,v + c l u // 0 c 1 (food for thought!)

26 Minimum Spanning Tree = A Greedy Problem Prim s algorithm. Start with T = a root node s, and greedily grow the tree T. At each step, add to T the cheapest edge e that has exactly one endpoint in T. Reverse-Delete algorithm. Start with T = E (all edges). Consider edges in descending order of cost. Delete edge e from T unless doing so disconnects T. Kruskal s algorithm. Start with T =. Consider edges in ascending order of cost. Insert edge e into T unless doing so creates a cycle. All three methods are greedy, and produce a minimum spanning tree.

27 Kruskal s Algorithm

28 From Spanning Trees to Steiner Trees The red dot is called a Steiner point (= an intermediate junction ) The maximum cost savings from adding Steiner points = min ratio of (min Steiner tree cost) / (min spanning tree cost) depends on the metric, or distance function Cf. Manhattan, Euclidean, Chebyshev, Adapted from Prof. G. Robins, UVA

29 Minimum Steiner Tree is a Hard Problem (Minimum Spanning Tree was Easy) You learn about hard vs. easy in CSE101, CSE105 How would you approach finding a low-cost Steiner tree? Adapted from Prof. G. Robins, UVA

30 Iterated 1-Steiner Algorithm (1990) Given a pointset S, what point p minimizes MST(S {p}) Algorithmic idea: Iterate! (greedily) In practice: solution cost is within 0.5% of OPT on average Adapted from Prof. G. Robins, UVA

31 Walks - Definition Let G = (V,E,φ) be a graph. A walk in G is a sequence: e 1, e,..., e n 1 of elements of E (edges of G) for which there is a sequence: a 1, a,..., a n of elements of V (vertices of G) such that φ(e i ) = {a i, a i+1 } for i = 1,,..., n 1.

32 Walks on Graphs d 1 b a c Which of the following are walks on the graph? A. a, b, c, d B. 1,,, C. d,b,a,c,d D. a,1,b E. None of the above / more than one of the above

33 Special Walks (Neither edges nor vertices need be distinct: walk) Edges must be distinct: trail Vertices must be distinct: path Sequence of vertices is the vertex sequence of the path Trail whose first vertex is same as last vertex: circuit Circuit with no repeated vertices (except start/end): cycle If the graph is directed: Obtain a directed walk, trail, path, circuit, cycle Replace {} with () in vertex set: φ(e i ) = (a i, a i+1 ) now Arrows: domain set on the left, codomain set on right

34 Example b 1 d e f c 6 a 5 Which of the following statements are true? A. There are no cycles in this graph. B. There are no circuits in this graph. C. There is a path that visits each vertex exactly once. D. There is a trail that uses each edge exactly once. E. None of the above/ more than one of the above.

35 More on graphs A graph is connected if there is a path between every pair of vertices. Edge case: graph with only one vertex, we say it is connected One vertex is reachable from another if there is a path between them. A connected component of a graph is a subset of vertices such that (i) every pair of vertices in the set is reachable from one another, and (ii) the subset is not part of a larger set with this property. Computing distance in a graph: breadth first search of all walks Small world phenomenon in social networks Analyze structure of the graph by looking at number of paths Hubs and spokes Link prediction in social networks

36 Visiting everywhere on a graph An Eulerian trail is a trail that includes every edge. An Eulerian circuit is a circuit that includes every edge.

37 Visiting everywhere on a graph An Eulerian trail is a trail that includes every edge. An Eulerian circuit is a circuit that includes every edge. In the previous example there is no Eulerian trail: b 1 d e f c 6 a 5 How do we prove this?

38 Visiting everywhere on a graph An Eulerian trail is a trail that includes every edge. An Eulerian circuit is a circuit that includes every edge. Observations: If a graph has an Eulerian circuit then it has an Eulerian trail. If a graph is not connected, then it does not have an Eulerian trail.

39 Characterization of Eulerian graphs Theorem. For a connected simple graph G, the following are equivalent: G contains an Eulerian circuit. Every vertex of G has even degree. The edges of G can be partitioned into edge-disjoint cycles. Corollary. A connected simple graph has an Eulerian trail if and only if the number of vertices with odd degree is either zero or two. UCSD connection: An Eulerian path approach to DNA fragment assembly Pavel A. Pevzner, Haixu Tang, and Michael S. Waterman (001).

40 Characterization of Eulerian graphs Theorem. For a connected simple graph G, the following are equivalent: G contains an Eulerian circuit. Every vertex of G has even degree. The edges of G can be partitioned into edge-disjoint cycles. Corollary. A connected simple graph has an Eulerian trail if and only if the number of vertices with odd degree is either zero or two. Application to previous example: 1 1 b a c d e f Four vertices with odd degree!

41 Characterization of Eulerian graphs Theorem. For a connected simple graph G, the following are equivalent: G contains an Eulerian circuit. Every vertex of G has even degree. The edges of G can be partitioned into edge-disjoint cycles. Corollary. A connected simple graph has an Eulerian trail if and only if the number of vertices with odd degree is either zero or two. Proof Idea: Eulerian circuits enter and leave vertices on distinct edges so there have to be an even number of edges incident with each vertex. If you can partition the graph into edge-disjoint cycles, you can string them together to get circuit.

42 Characterization of Eulerian graphs Theorem. For a connected simple graph G, the following are equivalent: G contains an Eulerian circuit. Every vertex of G has even degree. The edges of G can be partitioned into edge-disjoint cycles. Corollary. A connected simple graph has an Eulerian trail if and only if the number of vertices with odd degree is either zero or two. Tweaking the example: Two vertices with odd degree! 1 b a c e 6 f 5 1 1

43 Visiting everywhere on a graph, take A Hamiltonian cycle in a graph is a cycle that visits every vertex exactly once.

44 Visiting everywhere on a graph, take A Hamiltonian cycle in a graph is a cycle that visits every vertex exactly once. Example with an Eulerian trail does it have a Hamiltonian cycle? 1 b a c e 6 f 5 1 1

45 Visiting everywhere on a graph, take A Hamiltonian cycle in a graph is a cycle that visits every vertex exactly once. What about now does this graph have a Hamiltonian cycle? 1 b a c e f 6 g 5

46 Visiting everywhere on a graph, take A Hamiltonian cycle in a graph is a cycle that visits every vertex exactly once. A Hamiltonian graph is a graph that has a Hamiltonian cycle

47 Examples Which of the following is true: A. Both are Hamiltonian and Eulerian B. Neither is Hamiltonian and neither is Eulerian C. Both are Hamiltonian and not Eulerian D. One is Hamiltonian and not Eulerian; the other is Eulerian and not Hamiltonian E. None of the above.

48 Coloring a graph A proper coloring of a simple graph G = (V,E) is a function λ : V C, where C is a set of colors, such that {u,v} E λ(u) λ(v). There is a proper coloring of a complete graph on vertices using: A. 1 color B. colors (but there aren t any with 1 color) C. colors (but there aren t any with 1 or colors) D. colors (but there aren t any with 1 or or colors) E. None of the above.

49 Another Tree: -Coloring a Graph Three colors: R, G, B G B G Graph 5 R (from Lecture 11) G R B 1B,G By symmetry, can use any two colors for vertices 1 and R B Tree of Colorings G 1 B A graph is legally colored with k colors if each vertex has a color (label) between 1 and k (inclusive), and no adjacent vertices have the same color. 5R

50 Coloring a graph A proper coloring of a simple graph G = (V,E) is a function λ : V C, where C is a set of colors, such that {u,v} E λ(u) λ(v) There is a proper coloring of any bipartite graph using: A. 1 color B. colors (but there aren t any with 1 color) C. colors (but there aren t any with 1 or colors) D. colors (but there aren t any with 1 or or colors) E. None of the above.

51 Coloring problem and chromatic numbers The coloring problem: for any c >, devise an algorithm whose input can be any simple graph and whose output is the answer to the question can the graph be properly colored with c colors? Given G and c, deciding (yes/no) whether G can be properly colored with c colors is an NP-complete problem: no known fast algorithm to come up with solution; but easy to check if candidate solution works. The chromatic number of a graph G, (G), is the least number of colors needed to properly color G. In the previous example, we saw: (Complete graph, n vertices) = n, (Bipartite graph) = The Four Color Theorem says that every planar MAP can be properly colored with four colors. Does this mean that (any planar graph)? Planar Graph: can be drawn on a plane such that edges intersect only at endpoints

52 Hard graph problems we think Hamiltonian cycle problem: Given a simple graph G, decide whether G contains a Hamiltonian cycle. Coloring problem: Given graph G and c >, decide whether G can be properly colored with c colors. Traveling salesman problem: Given edge-weighted graph G and a number B, decide whether there is a Hamilton cycle through all vertices of G with total edge cost B. Clique problem: Given a simple graph G and a number s, decide whether there are s vertices in G with all C(s,) possible edges between them being present in G.

53 Problems 17 P17.1 Eleven people form five study groups. Prove that two students, A and B, can be found such that every study group that contains A also includes B. P17. In a game, Player 1 and Player take turns adding to the current number any natural number smaller than it. The game begins with the number. Whoever reaches the number 1000 wins. Which player has a winning strategy, and what is the winning strategy? P17. In a game, Player 1 and Player take turns subtracting from the current number any positive power of less than the current number. The game begins with the number Whoever reaches the number 0 wins. Which player has a winning strategy, and what is the winning strategy?