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Priority Models

Priority Models. Sashka Davis University of California, San Diego June 1, 2003. Goal. Define priority models, which are a formal framework of greedy algorithms Develop a technique for proving lower bounds for priority algorithms. The big picture.

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Priority Models

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  1. Priority Models Sashka Davis University of California, San Diego June 1, 2003

  2. Goal • Define priority models, which are a formal framework of greedy algorithms • Develop a technique for proving lower bounds for priority algorithms

  3. The big picture Classify the kinds of problems on which the different heuristics perform well Can we build a formal model for the different algorithmic design paradigms? Are the known algorithms optimal or can they be improved? Evaluate the limitations of each technique? Greedy Dynamic Programming Hill-climbing Polynomial Time Algorithms Divide and Conquer

  4. ShortPath ADAPTIVE FIXED Greedy heuristics • Priority algorithms are a formal model for greedy algorithms. PRIORITY ALGORITHMS

  5. Common structure of greedy algorithms • They sortitems (edges, intervals, etc.) • Consider each item once and either add it to the solution or throw it away

  6. Interval scheduling on a single machine • Instance: Set of intervals I=(i1, i2,…,in), j ij=[rj, dj] • Problem: schedule intervals on a single machine • Solution: S  I • Objective function: maxiS(dj - rj)

  7. A simple solution (LPT) Longest Processing Time algorithm input I=(i1, i2,…,in) • Initialize S ← • Sort the intervals in decreasing order (dj – rj) • while (I is not empty) • Let ik be the next in the sorted order • If ik can be scheduled then S ← S U {ik}; • I ← I \ {ik} • Output S

  8. LPT OPT OPT OPT LPT is a 3-approximation • LPT sorts the intervals in decreasing order according to their length • 3 LPT≥ OPT ri di

  9. The minimum cost spanning tree problem • Instance: Edge weighted graph • Problem: Find a tree of edges that spans V • Objective function Minimize the cost of the tree

  10. A solution for MST problem Kruskal’s algorithm Input (G=(V,E), w: E →R) • Initialize empty solution T • L = Sorted list of edges in increasing order according to their weight • while (L is not empty) • e = next edge in L • Add the edge to T, as long as T remains a forest and remove e from L • Output T

  11. Another solution to the MST problem Prim’s algorithm Input G=(V,E), w: E →R • Initialize an empty tree T ←; S ← • Pick a vertex u; S={u}; • for (i=1 to |V|-1) • (u,v) = min(u,v)cut(S, V-S)w(u,v) • S←S  {v}; T←T{(u,v)} • Output T

  12. Classification of the example algorithms LPT Kruskal Prim ADAPTIVE FIXED PRIORITY ALGORITHMS

  13. Talk outline • History of priority algorithms • Priority algorithm framework for scheduling problems • Priority algorithms for facility location • General framework of priority algorithms • Future research

  14. Results [BNR02] • Defined fixed and adaptive priority algorithms • Proved that fixed priority algorithms are less powerful than adaptive priority algorithms • Considered variety of scheduling problems and proved many non-trivial upper and lower bounds

  15. Results [AB02] Proved tight bounds on performance of • Adaptive priority algorithms for facility location in arbitrary spaces • Fixed priority algorithms for uniform metric facility location • Adaptive priority algorithms for set cover

  16. Results [DI02] • Defined a general model of priority algorithms • Proved a strong separation between the classes of fixed and adaptive priority algorithms • Proved a separation between the class of memoryless adaptive priority algorithms and adaptive priority algorithms with memory • Proved tight bound for performance of adaptivepriority algorithms for weighted vertex cover problem

  17. Talk outline • History of priority algorithms • Priority algorithm framework for scheduling problems [BNR02] • Priority algorithms for facility location • General framework of priority algorithms • Future research

  18. The defining characteristics of greedy algorithms [BNR02] • The order in which data items are considered is determined by a priority function, which orders all possible data items • The algorithm sees one input at a time • Decision made for each data item is irrevocable

  19. Fixed Priority Algorithms Adaptive Priority Algorithms Priority models • Difference: How the next data item is chosen

  20. Fixed priority algorithms [BNR02] Input: a set of jobs S Ordering: Determine without looking at S a total ordering of all possible jobs while (S is not empty) Jnext - next job in S according to the order above Decision: Make irrevocable decision for Jnext S ← S \ {Jnext }

  21. Adaptive priority algorithms [BNR02] Input: a set of jobs S while (not empty S) Ordering: Determine without looking at S a total ordering of all possible jobs Jnext - next job in S according to the order above Decision: Make irrevocable decision for Jnext S ← S \ {Jnext }

  22. Separation between fixed and adaptive priority algorithms [BNR02] • Theorem: No fixed priority algorithm can achieve an approximation ratio better than 3 for the interval scheduling problem on multiple machine configuration • Theorem: CHAIN-2 is an adaptive priority algorithm achieving an approximation ratio 2, for interval scheduling on a two machine configuration

  23. Online algorithms • Must service each request before the next request is received • Several alternatives in servicing each request • Online cost is determined by the options selected

  24. Connection between online and priority algorithms • Similarities • The instance is viewed one input at a time • Decision is irrevocable • Difference • The order of the data items

  25. Competitive analysis of online algorithms t=1; I=∅ • Round t • Adversary picks a data item t; I← I U {t} • Algorithm makes a decision σt for t: A←A U { (t, σt )} • Adversary chooses whether to end the game. If not, the next round begins: t←t+1 • Adversary picks a solution Bfor I offline • Algorithm is awarded payoff value(A) / value(B)

  26. Fixed priority game • Adversary selects a finite set of data items S0; I←;t ←1 • Algorithm picks a total order on S0 • Adversary restricts the remaining data items: S1 S0 Round t • Let t St be next data item in the order • Algorithm makes a decision σt for t : A←A U { (t, σt )} • Adversary restricts the set St+1 St – t; I ← I U {t}; • Adversary chooses whether to end the game. If not, the next round begins: t←t+1 • Adversary picks a solution Bfor I • Algorithm is awarded payoff value (A) / value (B)

  27. Example lower bound [BNR02] • Theorem1: No priority algorithm can achieve an approximation ratio better than 3 for the interval scheduling problem with proportional profit for a single machine configuration

  28. e q 2 q-1 q-1 1 3 2 1 3 Proof of Theorem 1 • Adversary’s move • Algorithm’s move: Algorithm selects an ordering • Let i be the interval with highest priority

  29. 2 1 3 i 2 k j 1 3 Adversary’s strategy • If Algorithm decides not to schedule i • During next round Adversary removes all remaining intervals and schedules interval i i

  30. 2 1 3 i j k i 2 k j 1 3 Adversary’s strategy • If i = and Algorithm schedules i • During next round the Adversary restricts the sequence: i

  31. 2 1 3 m 2 k j 1 3 Adversary’s strategy • If i = and Algorithm schedules i • During next round Adversary restricts the sequence: i m i

  32. Conclusion • Adversary can pick (q, e) so that the advantage gained is arbitrarily close to 3 • No priority algorithm (fixed or adaptive) can achieve an approximation ratio better than 3 • LPT achieves an approximation ratio 3 • LPT is optimal within the class of priority algorithms

  33. Talk outline • History of priority algorithms • Priority algorithm framework for scheduling problems • Priority algorithms for facility location [AB02] • General framework of priority algorithms • Future research

  34. [AB02] work on priority algorithms [AB02] proved lower bounds on performance of adaptive and fixed priority algorithms for the facility location problem in metric and arbitrary spaces, and the set cover problems

  35. [AB02] result • Theorem: No adaptive priority algorithm can achieve an approximation ratio better than log(n) for facility location in arbitrary spaces

  36. Adaptive priority game • Adversary selects a finite set of data items S0; I←;t ←1 • Round t • Algorithm picks a data item t and a decision σt for t : A←A U { (t, σt )} • Adversary restricts the set St+1 St – t; I ← I U {t}; • Adversary chooses whether to end the game, if not next round begins t← t+1 • Adversary picks a solution Bfor I • Algorithm is awarded payoff value(A)/value(B)

  37. Facility location problem • Instance is a set of cities and set of facilities • The set of cities is C={1,2,…,n} • Each facility fi has an opening cost cost(fi) and connection costs for each city: {ci1, ci2,…, cin} • Problem: open a collection of facilities such that each city is connected to at least one facility • Objective function: minimize the opening and connection costs min(ΣfScost(fi) + ΣjCmin fiScij )

  38. Adversary presents the instance: • Cities: C={1,2,…,n}, where n=2k • Facilities: • Each facility has opening cost n • City connection costs are 1 or∞ • Each facility covers exactly n/2 cities • cover(fj) = {i | i  C,cji=1} Cu denotes the set of cities not yet covered by the solution of the Algorithm

  39. Adversary’s strategy At the beginning of each round t • The Adversary chooses St to consist of facilities f such thatfStiff |cover(f)∩ Cu| = n/(2t) • The number of uncovered cities Cu is n/(2t-1) Two facilities are complementary if together they cover all cities in C. For any round t St consists of complementary facilities

  40. The game Uncovered cities Cu

  41. End of the game • Either Algorithm opened log(n) facilities or failed to produce a valid solution • Cost of Algorithm’s solution is n.log(n)+n • Adversary opens two facilities incurs total cost 2n+n

  42. Conlusion • The Adversary has a winning strategy • No adaptive priority algorithm can achieve an approximation ratio better than log(n)

  43. Talk outline • History of priority algorithms • Priority algorithm framework for scheduling problems • Priority algorithms for facility location • General framework of priority algorithms [DI02] • Future research

  44. Fixed priority algorithms [DI02] Input: instance ={1,2,…, n}, Output: solution 1. Determine an ordering function 2. Order I according to 3. Repeat • Let the next data item in the ordering be • Make a decision • Update the partial solution S until (decisions are made for all data items) 4. Output

  45. Adaptive priority algorithms [DI02] Input: instance ={1,2,…, n}, Output: solution 1. Initialization 2. Repeat • Determine an ordering function • Pick the highest priority data item according to • Make an irrevocable decision • Update: until (decisions are made for all data items; ) 3. Output

  46. Strong separation between fixed and adaptive priority algorithms • Theorem: No fixed priority algorithm can achieve any constant approximation ratio for the ShortPath problem • Dijkstra algorithm for the Single source shortest path problem solves the ShortPath problem exactly.

  47. The ShortPath problem Instance: Given an edge-weighted directed graph G=(V,E) and two nodes s and t Problem: Find a directed tree of edges, rooted at s Objective function: Minimize the combined weight of the edges on the path from s to t

  48. u(k) a y(1) v(1) s t z(1) x(1) b w(k) Adversary’s strategy

  49. v(1) w(k) Algorithm selects an order on S0 If then the Adversary presents: u(k) a y(1) s t x(1) z(1) b

  50. Adversary’s strategy • Wait until Algorithm considers edge Y(1). • Y(1) will be considered before Z(1) • Adversary can remove data items not yet considered

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