David Luebke 1 3/19/2016 CS 332: Algorithms Augmenting Data Structures.
David Luebke 1 9/7/2015 CS 332: Algorithms NP Complete: The Exciting Conclusion Review For Final.
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Transcript of David Luebke 1 9/7/2015 CS 332: Algorithms NP Complete: The Exciting Conclusion Review For Final.
David Luebke 2 04/19/23
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 3 04/19/23
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)
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David Luebke 4 04/19/23
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 5 04/19/23
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 6 04/19/23
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 7 04/19/23
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 PNP p Q
What does it mean if Q is NP-Complete? Q is NP-Hard and Q NP
David Luebke 8 04/19/23
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 R
is NP-Complete? A: Show R NP, and reduce a known
NP-Complete problem Q to R
David Luebke 9 04/19/23
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 v in 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 10 04/19/23
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 11 04/19/23
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 12 04/19/23
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 13 04/19/23
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 14 04/19/23
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 15 04/19/23
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 16 04/19/23
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 17 04/19/23
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 18 04/19/23
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 19 04/19/23
Clique Vertex Cover
First, show vertex cover in NP (How?) Next, reduce k-clique to vertex cover
The complement GC 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 20 04/19/23
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 21 04/19/23
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 22 04/19/23
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 23 04/19/23
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 24 04/19/23
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 25 04/19/23
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 26 04/19/23
Graph Representation
Adjacency list Adjacency matrix Tradeoffs:
What makes a graph dense? What makes a graph sparse? What about planar graphs?
David Luebke 27 04/19/23
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 28 04/19/23
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 29 04/19/23
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 30 04/19/23
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 31 04/19/23
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 32 04/19/23
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 33 04/19/23
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 34 04/19/23
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:
otherwise]),1[],1,[max(
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David Luebke 35 04/19/23
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