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CS 332: Algorithms

CS 332: Algorithms. NP Complete: The Exciting Conclusion Review For Final. Administrivia. Homework 5 due now All previous homeworks available after class Undergrad TAs still needed (before finals) Final exam Wednesday, December 13 9 AM - noon

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CS 332: Algorithms

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  1. CS 332: Algorithms NP Complete: The Exciting Conclusion Review For Final David Luebke 14/3/2014

  2. Administrivia • Homework 5 due now • All previous homeworks available after class • Undergrad TAs still needed (before finals) • Final exam • Wednesday, December 13 • 9 AM - noon • You are allowed two 8.5“ x 11“ cheat sheets • Both sides okay • Mechanical reproduction okay (sans microfiche) David Luebke 24/3/2014

  3. Homework 5 • Optimal substructure: • Given an optimal subset A of items, if remove item j, remaining subset A’ = A-{j} is optimal solution to knapsack problem (S’ = S-{j}, W’ = W - wj) • Key insight is figuring out a formula for c[i,w], value of soln for items 1..i and max weight w: • Time: O(nW) David Luebke 34/3/2014

  4. Review: P and NP • What do we mean when we say a problem is in P? • What do we mean when we say a problem is in NP? • What is the relation between P and NP? David Luebke 44/3/2014

  5. Review: P and NP • What do we mean when we say a problem is in P? • A: A solution can be found in polynomial time • What do we mean when we say a problem is in NP? • A: A solution can be verified in polynomial time • What is the relation between P and NP? • A: PNP, but no one knows whether P = NP David Luebke 54/3/2014

  6. Review: NP-Complete • What, intuitively, does it mean if we can reduce problem P to problem Q? • How do we reduce P to Q? • What does it mean if Q is NP-Hard? • What does it mean if Q is NP-Complete? David Luebke 64/3/2014

  7. Review: NP-Complete • What, intuitively, does it mean if we can reduce problem P to problem Q? • P is “no harder than” Q • How do we reduce P to Q? • Transform instances of P to instances of Q in polynomial time s.t. Q: “yes” iff P: “yes” • What does it mean if Q is NP-Hard? • Every problem PNP p Q • What does it mean if Q is NP-Complete? • Q is NP-Hard and Q  NP David Luebke 74/3/2014

  8. Review: Proving Problems NP-Complete • What was the first problem shown to be NP-Complete? • A: Boolean satisfiability (SAT), by Cook • How do we usually prove that a problem Ris NP-Complete? • A: Show R NP, and reduce a known NP-Complete problem Q to R David Luebke 84/3/2014

  9. Review: Directed  Undirected Ham. Cycle • Given: directed hamiltonian cycle is NP-Complete (draw the example) • Transform graph G = (V, E) into G’ = (V’, E’): • Every vertex vin V transforms into 3 vertices v1, v2, v3 in V’ with edges (v1,v2) and (v2,v3) in E’ • Every directed edge (v, w) in E transforms into the undirected edge (v3, w1) in E’ (draw it) David Luebke 94/3/2014

  10. Review:Directed  Undirected Ham. Cycle • Prove the transformation correct: • If G has directed hamiltonian cycle, G’ will have undirected cycle (straightforward) • If G’ has an undirected hamiltonian cycle, G will have a directed hamiltonian cycle • The three vertices that correspond to a vertex v in G must be traversed in order v1, v2, v3 or v3, v2, v1, since v2 cannot be reached from any other vertex in G’ • Since 1’s are connected to 3’s, the order is the same for all triples. Assume w.l.o.g. order is v1, v2, v3. • Then G has a corresponding directed hamiltonian cycle David Luebke 104/3/2014

  11. Review: Hamiltonian Cycle  TSP • The well-known traveling salesman problem: • Complete graph with cost c(i,j) from city i to city j •  a simple cycle over cities with cost < k ? • How can we prove the TSP is NP-Complete? • A: Prove TSP  NP; reduce the undirected hamiltonian cycle problem to TSP • TSP  NP: straightforward • Reduction: need to show that if we can solve TSP we can solve ham. cycle problem David Luebke 114/3/2014

  12. Review: Hamiltonian Cycle  TSP • To transform ham. cycle problem on graph G = (V,E) to TSP, create graph G’ = (V,E’): • G’ is a complete graph • Edges in E’ also in E have weight 0 • All other edges in E’ have weight 1 • TSP: is there a TSP on G’ with weight 0? • If G has a hamiltonian cycle, G’ has a cycle w/ weight 0 • If G’ has cycle w/ weight 0, every edge of that cycle has weight 0 and is thus in G. Thus G has a ham. cycle David Luebke 124/3/2014

  13. Review: Conjunctive Normal Form • 3-CNF is a useful NP-Complete problem: • Literal: an occurrence of a Boolean or its negation • A Boolean formula is in conjunctive normal form, or CNF, if it is an AND of clauses, each of which is an OR of literals • Ex: (x1  x2)  (x1  x3  x4)  (x5) • 3-CNF: each clause has exactly 3 distinct literals • Ex: (x1  x2  x3)  (x1  x3  x4)  (x5  x3  x4) • Notice: true if at least one literal in each clause is true David Luebke 134/3/2014

  14. 3-CNF  Clique • What is a clique of a graph G? • A: a subset of vertices fully connected to each other, i.e. a complete subgraph of G • The clique problem: how large is the maximum-size clique in a graph? • Can we turn this into a decision problem? • A: Yes, we call this the k-clique problem • Is the k-clique problem within NP? David Luebke 144/3/2014

  15. 3-CNF  Clique • What should the reduction do? • A: Transform a 3-CNF formula to a graph, for which a k-clique will exist (for some k) iff the 3-CNF formula is satisfiable David Luebke 154/3/2014

  16. 3-CNF  Clique • The reduction: • Let B = C1  C2  …  Ck be a 3-CNF formula with k clauses, each of which has 3 distinct literals • For each clause put a triple of vertices in the graph, one for each literal • Put an edge between two vertices if they are in different triples and their literals are consistent, meaning not each other’s negation • Run an example: B = (x  y  z)  (x  y  z )  (x  y  z ) David Luebke 164/3/2014

  17. 3-CNF  Clique • Prove the reduction works: • If B has a satisfying assignment, then each clause has at least one literal (vertex) that evaluates to 1 • Picking one such “true” literal from each clause gives a set V’ of k vertices. V’ is a clique (Why?) • If G has a clique V’ of size k, it must contain one vertex in each clique (Why?) • We can assign 1 to each literal corresponding with a vertex in V’, without fear of contradiction David Luebke 174/3/2014

  18. Clique  Vertex Cover • A vertex cover for a graph G is a set of vertices incident to every edge in G • The vertex cover problem: what is the minimum size vertex cover in G? • Restated as a decision problem: does a vertex cover of size k exist in G? • Thm 36.12: vertex cover is NP-Complete David Luebke 184/3/2014

  19. Clique  Vertex Cover • First, show vertex cover in NP (How?) • Next, reduce k-clique to vertex cover • The complementGC of a graph G contains exactly those edges not in G • Compute GC in polynomial time • G has a clique of size k iff GC has a vertex cover of size |V| - k David Luebke 194/3/2014

  20. Clique  Vertex Cover • Claim: If G has a clique of size k,GC has a vertex cover of size |V| - k • Let V’ be the k-clique • Then V - V’ is a vertex cover in GC • Let (u,v) be any edge in GC • Then u and v cannot both be in V’ (Why?) • Thus at least one of u or v is in V-V’ (why?), so edge (u, v) is covered by V-V’ • Since true for any edge in GC, V-V’ is a vertex cover David Luebke 204/3/2014

  21. Clique  Vertex Cover • Claim: If GC has a vertex cover V’  V, with |V’| = |V| - k, then G has a clique of size k • For all u,v V, if (u,v)  GC then u  V’ or v  V’ or both (Why?) • Contrapositive: if u  V’ and v  V’, then (u,v)  E • In other words, all vertices in V-V’ are connected by an edge, thus V-V’ is a clique • Since |V| - |V’| = k, the size of the clique is k David Luebke 214/3/2014

  22. General Comments • Literally hundreds of problems have been shown to be NP-Complete • Some reductions are profound, some are comparatively easy, many are easy once the key insight is given • You can expect a simple NP-Completeness proof on the final David Luebke 224/3/2014

  23. Other NP-Complete Problems • Subset-sum: Given a set of integers, does there exist a subset that adds up to some target T? • 0-1 knapsack: you know this one • Hamiltonian path: Obvious • Graph coloring: can a given graph be colored with k colors such that no adjacent vertices are the same color? • Etc… David Luebke 234/3/2014

  24. Final Exam • Coverage: 60% stuff since midterm, 40% stuff before midterm • Goal: doable in 2 hours • This review just covers material since the midterm review David Luebke 244/3/2014

  25. Final Exam: Study Tips • Study tips: • Study each lecture since the midterm • Study the homework and homework solutions • Study the midterm • Re-make your midterm cheat sheet • I recommend handwriting or typing it • Think about what you should have had on it the first time…cheat sheet is about identifying important concepts David Luebke 254/3/2014

  26. Graph Representation • Adjacency list • Adjacency matrix • Tradeoffs: • What makes a graph dense? • What makes a graph sparse? • What about planar graphs? David Luebke 264/3/2014

  27. Basic Graph Algorithms • Breadth-first search • What can we use BFS to calculate? • A: shortest-path distance to source vertex • Depth-first search • Tree edges, back edges, cross and forward edges • What can we use DFS for? • A: finding cycles, topological sort David Luebke 274/3/2014

  28. Topological Sort, MST • Topological sort • Examples: getting dressed, project dependency • What kind of graph do we do topological sort on? • Minimum spanning tree • Optimal substructure • Min edge theorem (enables greedy approach) David Luebke 284/3/2014

  29. MST Algorithms • Prim’s algorithm • What is the bottleneck in Prim’s algorithm? • A: priority queue operations • Kruskal’s algorithm • What is the bottleneck in Kruskal’s algorithm? • Answer: depends on disjoint-set implementation • As covered in class, disjoint-set union operations • As described in book, sorting the edges David Luebke 294/3/2014

  30. Single-Source Shortest Path • Optimal substructure • Key idea: relaxation of edges • What does the Bellman-Ford algorithm do? • What is the running time? • What does Dijkstra’s algorithm do? • What is the running time? • When does Dijkstra’s algorithm not apply? David Luebke 304/3/2014

  31. Disjoint-Set Union • We talked about representing sets as linked lists, every element stores pointer to list head • What is the cost of merging sets A and B? • A: O(max(|A|, |B|)) • What is the maximum cost of merging n 1-element sets into a single n-element set? • A: O(n2) • How did we improve this? By how much? • A: always copy smaller into larger: O(n lg n) David Luebke 314/3/2014

  32. Amortized Analysis • Idea: worst-case cost of an operation may overestimate its cost over course of algorithm • Goal: get a tighter amortized bound on its cost • Aggregate method: total cost of operation over course of algorithm divided by # operations • Example: disjoint-set union • Accounting method: “charge” a cost to each operation, accumulate unused cost in bank, never go negative • Example: dynamically-doubling arrays David Luebke 324/3/2014

  33. Dynamic Programming • Indications: optimal substructure, repeated subproblems • What is the difference between memoization and dynamic programming? • A: same basic idea, but: • Memoization: recursive algorithm, looking up subproblem solutions after computing once • Dynamic programming: build table of subproblem solutions bottom-up David Luebke 334/3/2014

  34. LCS Via Dynamic Programming • Longest common subsequence (LCS) problem: • Given two sequences x[1..m] and y[1..n], find the longest subsequence which occurs in both • Brute-force algorithm: 2m subsequences of x to check against n elements of y: O(n 2m) • Define c[i,j] = length of LCS of x[1..i], y[1..j] • Theorem: David Luebke 344/3/2014

  35. Greedy Algorithms • Indicators: • Optimal substructure • Greedy choice property: a locally optimal choice leads to a globally optimal solution • Example problems: • Activity selection: Set of activities, with start and end times. Maximize compatible set of activities. • Fractional knapsack: sort items by $/lb, then take items in sorted order • MST David Luebke 354/3/2014

  36. The End David Luebke 364/3/2014

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