Task Scheduling. A Lecture in CE Freshman Seminar Series: Ten Puzzling Problems in Computer Engineering. May 2016 Task Scheduling Slide 1

Transcription

1 Task Scheduling A Lecture in CE Freshman Seminar Series: Ten Puzzling Problems in Computer Engineering May 0 Task Scheduling Slide

2 About This Presentation This presentation belongs to the lecture series entitled Ten Puzzling Problems in Computer Engineering, devised for a ten-week, one-unit, freshman seminar course by Behrooz Parhami, Professor of Computer Engineering at University of California, Santa Barbara. The material can be used freely in teaching and other educational settings. Unauthorized uses, including any use for financial gain, are prohibited. Behrooz Parhami Edition Released Revised Revised Revised Revised First May 00 May 00 May 009 May 00 May 0 May 0 May 0 May 0 May 0 Task Scheduling Slide

3 Mini-Sudoku Puzzle Complete entries in this chart so that numbers - appear without repetition in each row, each column and each block Standard Sudoku consists of a 9 9 chart, but this mini version is good for a quick fix USA Today carries a daily mini-sudoku at its site: Sudoku isn t a math puzzle: We can use the letters A-F, or any other six symbols, instead of the numbers - May 0 Task Scheduling Slide

4 Mini-Sudoku Puzzle: Solution Method Complete entries in this chart so that letters A-F appear without repetition in each row, each column and each block AE C B D D C A To continue from here, write down all possible choices in the remaining blank boxes and see whether the resulting info leads to more progress F B D A C E A C D F SuDoKu: abbr. in Japanese for numbers must be single. Euler may have invented it; Howard Garns (US) & Wayne Gould (HK) popularized it in modern times D C F A AE C D May 0 Task Scheduling Slide

5 Sudoku Puzzle: Easy Example Complete entries in this chart so that numbers -9 appear without repetition in each row, each column and each block 9 Many newspapers carry these puzzles; there are also many collections in book form Sudoku puzzles of varying difficulties (easy, medium, hard, evil) are available at and several other Web sites, such as USA Today s site May 0 Task Scheduling Slide

6 Sudoku Puzzle Solution Method Strategy : When you can t make progress by Strategy, write down all candidate numbers in the cells and try to eliminate a number of options via reasoning. For example if xy, xy, xyz are candidates in three cells of a block, then the cell marked xyz must hold z Strategy : Identify a missing number from a row, column, or block; if you can exclude all but one cell for that number, then write it down ,,,,, 9 missing from this column missing from this row May 0 Task Scheduling Slide

7 Sudoku Puzzle: Hard Example Complete entries in this chart so that numbers -9 appear without repetition in each row, each column and each block Hard puzzles typically have fewer entries supplied, with each row, column, or block containing only a few entries Hard puzzles may have handles or starting points ( in the top left block or 9 in center and lower right blocks) May 0 Task Scheduling Slide

8 Constructing a (Mini-)Sudoku Puzzle Begin with a completed puzzle and one by one remove selected entries that can be deduced from the remaining ones This will ensure a unique solution, which is a desirable attribute Interesting fact: In a 9 9 Sudoku puzzle, you may need at least initial entries (clues) for the solution to be unique; no one knows whether a -clue puzzle with unique solution exists May 0 Task Scheduling Slide (Web page devoted to minimum Sudoku)

9 Variations on Sudoku Other sizes (e.g.,, with blocks; or, with blocks) Combining this 000s phenomenon with Rubik s cube of the 90s... or with the age-old sliding puzzle Latin square May 0 Task Scheduling Slide 9

10 Task Scheduling Problem We have a set of of tasks There are some processors that can execute tasks Assign tasks to processors so as to meet certain constraints A task may fit only some processors Tasks may have prerequisite tasks Preemption may be (dis)allowed Tasks may have deadlines Shortest schedule may be sought Numbers in Sudoku puzzle Cells in Sudoku puzzle can hold numbers Place numbers in cells while honoring some constraints Use only numbers -9 Some numbers already placed Different numbers in each row Different numbers in each column Different numbers in each block Virtually all instances of the task scheduling problem are difficult (NP-hard), just like Sudoku May 0 Task Scheduling Slide 0

11 Resource Allocation Problem We have a set of of resources There are locations where resources may be placed Assign resources to locations to meet certain constraints A resource may fit only some locations Resources must be easily accessible Resource mobility may be (dis)allowed Resource cost may differ by location Lowest-cost assignment may be sought Numbers in Sudoku puzzle Cells in Sudoku puzzle can hold numbers Place numbers in cells while honoring some constraints Use only numbers -9 Some numbers already placed Different numbers in each row Different numbers in each column Different numbers in each block Virtually all instances of the resource allocation problem are difficult (NP-hard), just like Sudoku May 0 Task Scheduling Slide

12 Scheduling Required CE Courses CS 0 Math A Math B ECE Phys Chem A Chem B Chem AL Chem BL Units units CS 0 Math C Phys 0 units CS 0 CS 0 Phys L Phys Math A ECE A Upper - division standing Engr 0 Phys L Phys ECE A ECE B ECE B ECE C Or CS 0 Constraints Prerequisite: Solid downward arrow Corequisite: Dashed sideways arrow Units per quarter: CS 0A CS 0 Or CS 0 Or PSTAT 0A ECE ECE A ECE B ECE 9 May 0 Task Scheduling Slide

13 Scheduling Required CE Courses CS 0 Math A Math B ECE Phys Chem A Chem B Chem AL Chem BL Units units 0 units CS 0 CS 0 CS 0 Math C Phys L Phys Phys Math A ECE A Upper - division standing Engr 0 Phys L Phys ECE A ECE B ECE B ECE C Or CS 0 Constraints Prerequisite: Solid downward arrow Corequisite: Dashed sideways arrow Units per quarter: CS 0A CS 0 Or CS 0 Or PSTAT 0A ECE ECE A ECE 9 ECE B Almost done! May 0 Task Scheduling Slide

14 Job Task Machine Time Staff Ja Ta M Ja Ta M Jb Tb M Jb Tb M Jb Tb M Jc Tc M Jd Td M Jd Td M M M M Job-Shop Scheduling S t a f f S t a f f Ta Tb Ta Td Td Tb Tc Tb Ta Tb Ta Td Tc Td Tb May 0 Task Scheduling Slide Tb Time Time

15 Schedule Refinement Job Task Machine Time Staff Ja Ta M Ja Ta M Jb Tb M Jb Tb M Jb Tb M Jc Tc M Jd Td M Jd Td M M M M S t a f f S t a f f Ta Ta Tb Tb Ta Ta Tb Td Tc Tb Td Switch? Td Tb Tc May 0 Task Scheduling Slide Tb Td Time Time

16 Truck Scheduling Seattle Portland Boise 0% Chicago SF % 0% % NYC LA Dallas 0% Orlando Truck location Required trip 00% Truck load May 0 Task Scheduling Slide

17 Multiprocessor Scheduling Task graph with unit-time tasks Here s a heuristic known as list scheduling:. Find the depth T of the task graph. Take T as a goal for the running time T p. Determine the latest possible start times. Assign priorities in order of latest times T = (execution time goal) Latest start times: see the layered diagram Priorities: shown on the diagram in red When two tasks have the same latest start time, a secondary tie-breaking rule is used Vertex v represents j j Task task or Computation computation j May 0 Task Scheduling Slide x 0 x 0 y T p T Latency p Latency with with p proce p T T Number Number of nodes of nodes (h T T Depth Depth of the of the graph ( x Output Output 9 9

18 Assignment to Processors Tasks listed in priority order * 9 0 Vertex v represents j j Task task or Computation computation j x x T p T Latency p Latency with with p proce p T T Number Number of nodes of nodes (h T T Depth Depth of the of the graph ( P 9 0 x P P P P P Even in this simple case of unit-time tasks, multiprocessor scheduling remains difficult with as few as processors Time Step 0 0 y Output Output 9 9 May 0 Task Scheduling Slide

19 Two Related and Similar Problems The knapsack problem We have storage capacity W and n files of sizes w i and values v i Pick a max-value subset of files that fit in the storage space W Files cannot be broken into pieces Naïve solution: Examine all n subsets Dynamic programming solution Various heuristic aids Approx. solutions (say, 90% of optimal) Off-line game of Tetris We have a rectangular bin and a sequence of tetrominos Find optimal play to maximize the number of pieces used Pieces can only be rotated Exponentially many choices There are many other related and similarly hard problems, some of which don t even admit efficient approximations May 0 Task Scheduling Slide 9