Transcript 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)
– 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
Graham’s Scan (35.3)
O(nlogn)
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.
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
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)
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
NP-Completeness (36.4-5)
• P: yes and no in pt
• NP: yes in pt
• NPH  NPC
NP-hard
NPC
P
NP
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
x’
a’
z
z’
x
y’
a
y
x, z’, y independent
F is satisfiable:
f = 1 if x = z’ = y = 1
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
red independent set
G
noedge  edge
Complement G’
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
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
red elements of ground set X
a
b
c
blue subsets in family F
A
B
A = {a,b,c}, B = {c,d}
d
Intermediate Classes
NP-hard
NPC
P
 NP 
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
Runtime Complexity Classes
• Runtime order:
• Example
– constant
– adding an element in a queue/stack
– almost constant
– inverse Ackerman function = O(loglog…log n)
n times
– logarithmic
– extracting minimum from binary heap
– sublinear
– n1/2
– linear
– traversing binary search tree, list
– pseudolinear
– quadratic
– O(n log n) sorting n numbers, closest pair,
MST, Dijkstra shortest paths
– polynomial
– adding two nn matrices
– subexponential
–
– exponential
– e n, , n!
– superexponential
– Ackerman function
e n ^ (1/2)
.
.
2
2
.
2
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
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)}
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
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
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
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
1
k+2
2
3
1
… 2k-1
k+3
4
…k
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
Steiner Tree Problem
Given: A set S of points in the plane = terminals
Find: Minimum-cost tree spanning S = minimum Steiner tree
Euclidean metric
Terminals
1
1
1
Cost = 2
1
Cost = 3
Steiner Point
1
Rectilinear metric
Cost = 6
1
1
Cost = 4
1
1
1
1
1
1
1
1
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)
MST -Heuristic
Theorem: MST-heuristic is a 2-approximation in graphs
Proof: MST < Shortcut Tour  Tour = 2 • OPTIMUM
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
2
3
MST Cost = 2k-2
Steiner Point
Opt Cost = k
Approximation ratio = 2-2/k  2
4
k
5
The Set Cover Problem
• Sets Ai cover 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
Greedy Algorithm for SCP
1
2
6
3
4
5
• Greedy Algorithm:
– While X is not empty
• find s  F minimizing w(s) / |s  X|
• X =X-s
• C=C+s
– Return C
Analysis of Greedy Algorithm
• Th: APR of the Greedy Algorithm is at most 1+ln k
• Proof:
ki | Si 1 |
| Si  X i | ni
ki  ki 1  ni 1
wi opt

ni
ki

wi
wi 

ki  ki 1 
ki 1  ki 1 1 
opt
 opt 

k0
wi 
wi
  
ln
 ln  1 
klast
opt
 opt 
wi ki
ni 
opt

wi 

klast  k0   1 
 opt 
ln( 1  x)  x
k0
approx
1  ln

klast
opt
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