Lecture-8-CS345A-2023 of Design and Analysis
- 1. Design and Analysis of Algorithms Lecture 8 • Job scheduling problem (continued) • Synchronizing a circuit with minimum delay 1 Algorithms-II : CS345A
- 2. A Job scheduling problem Recap the last lecture 2
- 3. • There are jobs: • Each job takes certain time for execution. • Each job also has a deadline. • There is a single server. All jobs need to be scheduled. Aim: Compute an order , ,…, in which the jobs should be scheduled such that maximum lateness is minimized. 3 , ,…, Job takes time Job has deadline time 𝑗𝑖 1 𝑗𝑖 𝑛 𝑗𝑖 2 𝑗𝑖 3 𝟎 𝑗𝑖 𝑘 𝑑𝑖 𝑘 𝑓 𝑖𝑘 𝑑𝑖 𝑘 Lateness of
- 4. Scheduling 2 jobs Lemma:(for 2 jobs) Scheduling ``the earlier deadline job’’ an optimal solution 4 𝑗1 𝑗 2 𝑡 2 𝑡1 𝑑2 𝑑1 time 𝑗1 𝑗 2 𝟎 In the last class, we realized Time of execution is immaterial
- 5. How to schedule more than 2 jobs ? Lemma:(for 2 jobs) Scheduling ``the earlier deadline job’’ an optimal solution 5 time 𝟎 Consider any schedule
- 6. How to schedule more than 2 jobs ? Lemma:(for 2 jobs) Scheduling ``the earlier deadline job’’ an optimal solution • Schedule the jobs in increasing order of their deadlines. 6 𝟎 time ``jobs scheduled in the increasing order of deadline’’ is an optimal schedule Algorithm: time Consider any schedule Consider any pair of consecutive jobs. If they are not scheduled in the increasing order of their deadlines, swapping them will only minimize the maximum lateness of the schedule By repeating this step for each pair of consecutive jobs, we can transform the schedule into a schedule whose maximum lateness is less than or equal to that of the original schedule. How do the jobs appear in the final schedule ? In increasing order of deadlines. What if the original schedule was an optimal schedule ?
- 7. SYNCHRONIZING A CIRCUIT with minimum delay enhancement 7
- 8. An Electric Circuit Electric signal originates at the root Signal passing through each edge incurs a delay (a few nano seconds) Finally signal reaches all the leaves. 8 5 6 2 2 3 2 6 2 1 2 1 4 2 3 Delay 11 10 14 10 10 11 10 9 Electric signal A complete binary tree 14 10
- 9. Problem definition Given: • There is a circuit in the form of a complete binary tree. • Electric signal propagates from root to all leaf nodes. • Each edge has certain delay • The delay in reaching signal to a leaf node = “sum of delays on all edges on the path from root.” Objective: Enhance delay along certain edges so that • The delay on all paths from root to leaf nodes is the same. • Total delay enhancement is minimum. 9 What is the first step for designing an algorithm ?
- 10. Working on an Example 10
- 11. Working on an Example Total delay enhancement: 27 11 5 6 2 2 3 2 6 2 1 2 1 4 2 3 Delay 11 10 14 10 10 11 10 9 Electric signal +3 +4 +4 +3 +4 +5 New Delay 14 14 14 14 14 14 14 14 How to achieve 1st Objective: Synchronizing all paths from the root? How much enhancement is needed for this edge ? +4 is necessary What algorithm comes to your mind based on this inference ? Is this the minimum delay enhancement ? +4 Spend some time on this portion to see if you can reduce the total delay ?
- 12. Working on an Example Total delay enhancement: 27 12 5 6 2 2 3 2 6 2 1 2 1 4 2 3 Delay 11 10 14 10 10 11 10 9 Electric signal +3 +4 +4 +4 +3 +1 New Delay 14 14 14 14 14 14 14 14 +4 23 What algorithm comes to your mind based on this inference ? 𝒖 +1 is necessary for . 𝒗 Recall from the past slide +4 is necessary for .
- 13. Toward designing an algorithm Total delay enhancement: 14 13 5 6 2 2 3 2 6 2 1 2 1 4 2 3 Delay 11 10 14 10 10 11 10 9 Electric signal +1 +1 +4 +1 +3 +3 +1 10 11 14 11 11 11 14 14 14 14 14 14 For nodes just above the leaf level What about the nodes at higher levels ? All this enhancement was absolutely necessary. Inspired by the processing done at the lower levels, the following seems to be the way to process higher nodes.
- 14. Overview of the proposed algorithm Process the nodes in bottom up order. A non-leaf node is processed as follows: • Compute the max-delay path from to a leaf node lying in its left subtree. • Compute the max-delay path from to a leaf node lying in its right subtree. If the two delays differ, “enhance the delay of one of its edges accordingly” 14 We need to describe this step more formally.
- 15. u For each non-leaf node do the following: : max delay along any leftward path : max delay along any rightward path If () increase delay of right edge by - Else increase delay of left edge by - 15 𝜷 𝜶 𝜸 𝑫𝑹(𝒖) 𝑫𝑳(𝒖) +𝑫𝑳(𝒖) − 𝑫𝑹 (𝒖) +𝑫𝑹 (𝒖) − 𝑫𝑳(𝒖)
- 16. PROOF OF CORRECTNESS OF THE ALGORITHM 16 WHAT ASSERTION SUFFICES AS A PROOF ?
- 17. What assertion suffices as a proof ? Usually it is difficult even to find out the claim whose establishment captures the correctness of the algorithm. In the current algorithm, what might be this claim ? Let us have a re-look at the algorithm from point of view of a single node . 17
- 18. u For each non-leaf node do the following: : max delay along any leftward path : max delay along any rightward path If () increase delay of right edge by - Else increase delay of left edge by - Delay enhancement by : | - | 18 𝜷 𝜶 𝜸 𝑫𝑹(𝒖) 𝑫𝑳(𝒖) +𝑫𝑳(𝒖) − 𝑫𝑹 (𝒖)
- 19. Claim : There is an optimal solution where, the delay enhancement by = || Question: How to prove the claim ? Answer: make careful observations about the algorithm . 19
- 20. Observation 1 Root is synchronized in the optimal solution. Question: What can we say about the synchronization of any node in the optimal solution? Answer: Every node must be synchronized. (proof by contradiction: append the path from root to to get two paths with different delays.) 20 Electric signal 𝜹𝟏 𝜹𝟐 𝒖
- 21. Observation 2 Question: In optimal sol., what is the max. delay along any path from to a leaf node ? Guess: It remain unchanged. In other words, it will still be max(). 21 5 6 2 2 3 2 6 2 1 2 1 4 2 3 Delay 11 10 14 10 10 11 10 9 Electric signal 𝒖 +1 +1 +4 +1 +3 +3 +1 New Delay 14 14 14 14 14 14 14 14 The maximum delay from the root to any leaf node remains unchanged in the optimal solution. What can we say about the maximum delay from any other node to any leaf node in its subtree?
- 22. Usefulness of the guess 22
- 23. u Usefulness of the guess In the optimal solution, we must increase delay of right edge by ?? Delay enhancement by : | - | Guess proof of correctness of algorithm 23 𝜷 𝜶 𝜸 Suppose 𝑫𝑹(𝒖) 𝑫𝑳(𝒖) +𝑫𝑳(𝒖) − 𝑫𝑹 (𝒖) - unchanged unchanged How much enhancement ? 0 How much enhancement ?
- 24. Proving the guess Guess : In the optimal solution, the delay along any path from to any leaf node is max(). Question: How to prove it ? Answer: By contradiction. 24
- 25. What if the assertion fails at many nodes ? Homework: Make sincere attempt to get proof by contradiction. 25