An Analysis of First-Fit N.S. Narayanaswamy (IITM) Work with R. Subhash Babu

Interval Graphs Clique number ω: The maximum no. of intervals that share a point.

Coloring Intervals Resource Allocation: Each interval ~ Request for a resource Each interval ~ Request for a resource for a period of time for a period of time Color ~ Resource Color ~ Resource time color

Coloring Offline coloring Offline coloring Optimal coloring : Consider intervals in non-decreasing order of their left end points, assign the least feasible color. Optimal coloring : Consider intervals in non-decreasing order of their left end points, assign the least feasible color. chromatic number = clique number chromatic number = clique number Online Coloring Online Coloring Requests/intervals are presented in a sequences one after another. Requests/intervals are presented in a sequences one after another. Example

Competitive Ratio of First-Fit Competitive Ratio of A No of colors used by the online A No of colors used by the optimal offline algorithm = max G,s = No of colors used by First-Fit ω

First fit Principle: Assign the least feasible color to the incoming interval. Principle: Assign the least feasible color to the incoming interval. Currently proved to use at most 8ω-3 colors (our work). Currently proved to use at most 8ω-3 colors (our work). First 8ω by Kierstead, Brightwell, Trotter improving 10ω by Pemmaraju, Raman, Varadarajan. First 8ω by Kierstead, Brightwell, Trotter improving 10ω by Pemmaraju, Raman, Varadarajan. There exists an instance on which First fit uses at least 4.4ω colors. There exists an instance on which First fit uses at least 4.4ω colors.

First fit: Example Clique size is 2, therefore offline uses 2. Clique size is 2, therefore offline uses 2. No of colors used is 4 No of colors used is 4

Properties of First fit Property: If an interval I is colored j, then there exists an interval I’ in each color i, 1 ≤ i ≤ j such that I intersects I’. Property: If an interval I is colored j, then there exists an interval I’ in each color i, 1 ≤ i ≤ j such that I intersects I’. Wall like structure, height=number of colors used Wall like structure, height=number of colors used

Analysis of First-Fit Goal : Find a lower bound on the clique size in terms of the height of the wall. Goal : Find a lower bound on the clique size in terms of the height of the wall. Column construction procedure : a counting technique Column construction procedure : a counting technique Idea Idea Consider the First-Fit wall as a grid. Consider the First-Fit wall as a grid. Assign one of three symbols to each cell in the grid. Assign one of three symbols to each cell in the grid. Count relative occurrences of symbols. Count relative occurrences of symbols. Lower bound the clique size Lower bound the clique size

Column Construction Procedure m C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4 Elementary Columns/Intervals Elementary Columns/Intervals Assign symbols to each cell – R, $, F ( , ,  ) Assign symbols to each cell – R, $, F ( , ,  ) Find relations between ,  and  to find a lower bound on the clique number. Find relations between ,  and  to find a lower bound on the clique number.

Column Construction RRRRRRRRR C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4 First Step: Consider C 1 = set of cells colored 1, they get the label R. First Step: Consider C 1 = set of cells colored 1, they get the label R. Ending Condition: For i = 2,3,… stop when C i obtained from C i-1 becomes empty. Ending Condition: For i = 2,3,… stop when C i obtained from C i-1 becomes empty.

Rule - R RRRRRRRRR C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4 RRRR R1: For each cell e in C i-1, if e is occupied by an interval I colored i, then add e to C i with the symbol R.

Rule - $ RRRRRRRRR C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4 RRRR R2: For each remaining cell e є C i-1, if e has a neighbor e’ in C i-1 which is added to C i by rule R1, then add e to C i with the symbol $. $$$$

Rule - F e e l e r i-1 j j R3 : For each remaining cell e in C i-1, if e has a neighbor e' in C i-1 and e' is neighbor of e down to the level j, and  e (j, i) > (i - j)/, then add e to C i with the label “F”. All other columns become inactive Rules are applied in the order R1, R2 and R3.

Final Picture RRRRRRRRR C1C1C1C1 C2C2C2C2 C3C3C3C3 C4C4C4C4 RRRR$$$$ R F $ F RR RR $FFFF FFFF$$$ FFFFF m’

Analysis of the Height m ≤ m’. m ≤ m’. For each i ≤ j, an interval colored j intersects C i For each i ≤ j, an interval colored j intersects C i For 1 ≤ i ≤ m’ and e є C i, For 1 ≤ i ≤ m’ and e є C i,  e (i) ≥ (m -  e (i)) /  e (i) ≥ (m -  e (i)) / Proof by induction on i Proof by induction on i  e (m') ≤ 2m'/  e (m') ≤ 2m'/ m’ crucial here. Without this there was a weaker upper bound, and hence 10 competitiveness. m’ crucial here. Without this there was a weaker upper bound, and hence 10 competitiveness.

Proof Outline e i1i1 i2i2 i3i3 j1j1 j2j2 j3j3 j4j4   e (m') ≤ 2m'/  ⇒  e (m') +  e (m') ≥ m' – 2m'/  ⇒  e (m') ≥ m'/ (1 -2/ )  ⇒  e (m) ≥ m/ (1 - 2/ )  =4 ⇒  e (m) ≥ m/8, for =4

18 New Observations C 1 needs only to be cells corresponding to a minimal clique cover of the underlying interval graph. C 1 needs only to be cells corresponding to a minimal clique cover of the underlying interval graph. Analysis works by changing the order of rules to R1, R3, and R2. This yields a nicer proof of Analysis works by changing the order of rules to R1, R3, and R2. This yields a nicer proof of For 1 ≤ i ≤ m’ and e є C i,  e (i) ≥ (m -  e (i)) / For 1 ≤ i ≤ m’ and e є C i,  e (i) ≥ (m -  e (i)) / Thank You

Online Vs Offline Offline: Online: (1,3,2,4) [First fit] Back

1-Wall: Examples 1-wall Back

2-Wall: Examples 2-wall Back

3-Wall: Examples 3-wall Back