1 / 28

Convex Hull (35.3)

Convex Hull (35.3). Convex Hull, CH(X), is the smallest convex polygon containing all points from X, |X|=n Different methods: incremental: moving from left to right updates CH(x1..xi). the runtime O(nlogn)

roanna-moon
Download Presentation

Convex Hull (35.3)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Convex Hull(35.3) • Convex Hull, CH(X), is the smallest convex polygon containing all points from X, |X|=n • Different methods: • incremental: moving from left to right updates CH(x1..xi). the runtime O(nlogn) • divide-and-conquer divides into two subsets left and right in O(n), then combine into one convex hull in O(n) • prune-and-search O(n logh), where h is the # points in CH uses pruning as for finding median to find upper chain

  2. Graham’s Scan (35.3) O(nlogn)

  3. Finding the Closest Pair(35.4) • Brute-force: O(n2) • Divide-and-conquer algorithm with recurrence T(n)=2T(n/2)+O(n) • Divide: divide into almost equal parts by a vertical line which divides given x-sorted array X into 2 sorted subarrays • Conquer: Recursively find the closest pair in each half of X. Let  = min{left, right} • Combine: The closest pair is either in distance  or a pair of points from different halves.

  4. Combine in D-a-C (35.4) • Subarray Y’ (y-sorted) of Y with points in 2 strip • pY’ find all in Y’ which are closer than in  • no more than 8 in   2 rectangle • no more than 7 points can be closer than in  • If the closest in the strip closer then it is the answer 2 2   left right

  5. Voronoi Graph • Voronoi region Vor(p) (p in set S) • the set of points on the plane that are closer to p than to any othe rpoint in S • Voronoi Graph VOR(S) • dual to voronoi region graph • two points are adjacent if their voronoi regions have common contiguous boundary (segment)

  6. Voronoi Graph • Voronoi Graph in the rectilinear plane • Rectilinear distance: p = (x, y); p’=(x’,y’) Voronoi region of b ab b a bc c ac

  7. NP-Completeness (36.4-5) • P: yes and no in pt • NP: yes in pt • NPH  NPC NP-hard NPC P NP

  8. x x’ a’ z z’ x y’ a y Independent Set • Independent set in a graph G: pairwise nonadjacent vertices • Max Independent Set is NPC • Is there independent set of size k? • Construct a graph G: • literal -> vertex • two vertices are adjacent iff • they are in the same clause • they are negations of each other • 3-CNF with k clauses is satisfiable iff G has independent k-set • assign 1’s to literals-vertices of independent set • Example: f = (x+z+y’) & (x’+z’+a) & (a’+x+y) x, z’, y independent F is satisfiable: f = 1 if x = z’ = y = 1

  9. red independent set MAX Clique • Max Clique (MC): • Find the maximum number of pairwise adjacent vertices • MC is in NP • for the answer yes there is certificate of polynomial length = clique which can be checked in polynomial time • MC is in NPC • Polynomial time reduction from MIS: • For any graph G any independent set in G 1-1 corresponds to clique in the complement graph G’ red clique noedge  edge edge  noedge G Complement G’

  10. Minimum Vertex Cover • Vertex Cover: • the set of vertices which has at least one endpoint in each edge • Minimum Vertex Cover (MVC): • the set of vertices which has at least one endpoint in each edge • MIN Vertex Cover is NPC • If C is vertex cover, then V - C is an independent set red independent set blue vertex cover

  11. Set Cover • Given: a set X and a family F of subsets of X, F  2X, s.t. X covered by F • Find : subfamily G of F such that G covers X and |G| is minimize • Set Cover is NPC • reduction from Vertex Covert • Graph representation: edge between set A F and element x  X means x  A d a b c red elements of ground set X blue subsets in family F A B A = {a,b,c}, B = {c,d}

  12. NP-hard NPC P  NP  Intermediate Classes Dense Set Cover is NP but not in P neither in NPC Dense Set Cover: Each element of X belongs to at least half of all sets in F

  13. Runtime Complexity Classes • Example • adding an element in a queue/stack • inverse Ackerman function = O(loglog…log n) n times • extracting minimum from binary heap • n1/2 • traversing binary search tree, list • O(n log n) sorting n numbers, closest pair, MST, Dijkstra shortest paths • adding two nn matrices • en ^ (1/2) • en, , n! • Ackerman function • Runtime order: • constant • almost constant • logarithmic • sublinear • linear • pseudolinear • quadratic • polynomial • subexponential • exponential • superexponential 2 . . . 2 2 n times

  14. Hamiltonian Cycle and TSP • Hamiltonian Cycle: • given an undirected graph G • find a tour which visits each point exactly once • Traveling Salesperson Problem • given a positive weighted undirected graph G (with triangle inequality = can make shortcuts) • find a shortest tour which visits all the vertices • HC and TSP are NPC • NPC problems: SP, ISP, MCP, VCP, SCP, HC, TSP

  15. Approximation Algorithms (37.0) • When problem is in NPC try to find approximate solution in polynomial-time • Performance Bound = Approximation Ratio (APR) (worst-case performance) • Let I be an instance of a minimization problem • Let OPT(I) be cost of the minimum solution for instance I • Let ALG(I) be cost of solution for instance I given by approximate algorithm ALG APR(ALG) = max I {ALG(I) / OPT(I)} • APR for maximization problem = max I {ALG(I) / OPT(I)}

  16. Vertex Cover Problem (37.1) • Find the least number of vertices covering all edges • Greedy Algorithm: • while there are edges • add the vertex of maximum degree • delete all covered edges • 2-VC Algorithm: • while there are edges • add the both ends of an edge • delete all covered edges • APR of 2-VC is at most 2 • e1, e2, ..., ek - edges chosen by 2-VC • the optimal vertex cover has 1 endpoint of ei • 2-VC outputs 2k vertices while optimum  k

  17. 2-approximation TSP (37.2) • Given a graph G with positive weights Find a shortest tour which visits all vertices • Triangle inequality w(a,b) + w(b,c)  w(a,c) • 2-MST algorithm: • Find the minimum spanning tree MST(G) • Take MST(G) twice: T = 2  MST(G) • The graph T is Eulerian - we can traverse it visiting each edge exactly once • Make shortcuts • APR of 2-MST is at most 2 • MST weight  weight of optimum tour • any tour is a spanning tree, MST is the minimum

  18. 3/2-approximation TSP (Manber) • Matching Problem (in P) • given weighted complete (all edges) graph with even # vertecies • find a matching (pairwise disjoint edges) of minimum weight • Christofides’s Algorithm (ChA) • find MST(G) • for odd degree vertices find minimum matching M • output shortcutted T = MST(G) + M • APR of ChA is at most 3/2 • |MST|  OPT • |M| OPT/2 • |T|  (3/2) OPT odd

  19. 3/2-approximation TSP • Christofides’s Algorithm (ChA) • find MST(G) • for odd degree vertices find minimum matching M • output shortcutted T = MST(G) + M • The worst case for Christofides heuristic in Euclidean plane: - Minimum Spanning Tree length = 2k - 2 - Minimum Matching of 2 odd degree nodes = k - 1 - Christofides heuristic length = 3k - 3 - Optimal tour length = 2k - 1 - Approximation Ratio of Christofides = 3/2-1/(k-1/2) 1 k+1 k+2 k+3 … 2k-1 1 1 2 3 4 … k 1

  20. Non-approximable TSP (37.2) • Approximating TSP w/o triangle inequality is NPC • any c-approximation algorithm can solve Hamiltonian Cycle Problem in polynomial time • Take an instance of HCP = graph G • Assign weight 0 to any edge of G • Complete G up to complete graph G’ • Assign weight 1 to each new edge • c-approximate tour can use only 0-edges - so it gives Hamiltonian cycle of G

  21. Euclidean metric Steiner Point Cost = 2 1 1 Cost = 3 Terminals 1 1 1 Rectilinear metric Cost = 6 Cost = 4 1 1 1 1 1 1 1 1 1 1 Steiner Tree Problem Given: A set S of points in the plane = terminals Find:Minimum-cost tree spanning S = minimum Steiner tree

  22. Steiner Tree Problem in Graphs • Given a graph G=(V,E,cost) and terminals S in V Find minimum-cost tree spanning all terminals • MST algorithm (does not use Steiner points): • find G(S) = complete graph on terminals • edge cost = shortest path cost • find T(S) = MST of G(S) • replace each edge of T(S) with the path in G • output T(S)

  23. MST -Heuristic Theorem: MST-heuristic is a 2-approximation in graphs Proof: MST<Shortcut TourTour = 2 • OPTIMUM

  24. Approximation Ratios • Euclidean Steiner Tree Problem • approximation ratio = 2/3 • Rectilinear Steiner Tree Problem • approximation ratio = 3/2 • Steiner Tree Problem in graphs • approximation ratio = 2 1 MST Cost = 2k-2 2 3 Opt Cost = k Steiner Point Approximation ratio = 2-2/k  2 4 k 5

  25. The Set Cover Problem • Sets Aicover a set X if X is a union of Ai • Weighted Set Cover Problem Given: • A finite set X (the ground set X) • A family of F of subsets of X, with weights w: F  + Find: • sets S  F, such that • S covers X, X = {s | s  S} and • S has the minimum total weight  {w(s) | s  S} • If w(s) =1 (unweighted), then minimum # of sets

  26. Greedy Algorithm for SCP 1 • Greedy Algorithm: • While X is not empty • find s  F minimizing w(s) / |s  X| • X = X - s • C = C + s • Return C 2 6 3 4 5

  27. Analysis of Greedy Algorithm • Th: APR of the Greedy Algorithm is at most 1+ln k • Proof:

  28. Approximation Complexity • Approximation algorithm = polynomial time approximation algorithm • PTAS = a series of approximation algorithms s.t. for any  > 0 there is pt (1+)-approximation • There is PTAS fro subset sum • Remarkable progress in 90’s (assuming P  NP). • No PTAS for Vertex Cover • No clog k-approximation for Set Cover for k < 1 • k is the size of the ground set X • No n1- approximation for Independent Set • n is the number of vertices

More Related