CS 332: Algorithms

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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

NP Complete: The Exciting Conclusion

Review For Final

David Luebke 14/3/2014

• 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• 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

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

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

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

Graph Representation
• What makes a graph dense?
• What makes a graph sparse?

David Luebke 264/3/2014

Basic Graph Algorithms
• 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

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

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

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

Disjoint-Set Union
• 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

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

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

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

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

The End

David Luebke 364/3/2014