Greedy_Algorithm_Detailed_Presentation_in_DAA.pptx

Greedy_Algorithm_Detailed_Presentation_in_DAA.pptx

• 1.
  GREEDY ALGORITHMS DETAILED EXPLANATIONWITH EXAMPLES
• 2.
  GREEDY ALGORITHM • APROBLEM-SOLVING APPROACH THAT BUILDS A SOLUTION STEP BY STEP. • AT EACH STEP, THE ALGORITHM PICKS THE BEST AVAILABLE OPTION. • WORKS WHEN LOCAL CHOICES LEAD TO A GLOBALLY BEST SOLUTION. • FASTER AND SIMPLER THAN DYNAMIC PROGRAMMING IN MANY CASES. • EXAMPLES: ACTIVITY SELECTION, HUFFMAN CODING, KRUSKAL’S MST.
• 3.
  KEY FEATURES • GREEDY-CHOICEPROPERTY: CHOOSING THE LOCALLY BEST IS SAFE. • OPTIMAL SUBSTRUCTURE: OPTIMAL SOLUTION CAN BE BUILT FROM OPTIMAL SUBPROBLEMS. • EFFICIENT: OFTEN WORKS IN O(N LOG N) OR O(N). • NOT ALWAYS CORRECT: ONLY WORKS ON PROBLEMS THAT FIT GREEDY STRUCTURE.
• 4.
  ACTIVITY SELECTION PROBLEM •WE HAVE ACTIVITIES WITH START AND FINISH TIMES. • GOAL: SELECT MAXIMUM ACTIVITIES WITHOUT OVERLAP. • EXAMPLE: • ACTIVITIES: (1,4), (3,5), (0,6), (5,7), (8,9). • GREEDY SOLUTION: PICK EARLIEST FINISHING (1,4), (5,7), (8,9). →
• 5.
  WHY GREEDY WORKS? •GREEDY-CHOICE PROPERTY HOLDS: PICKING EARLIEST FINISHING LEAVES SPACE FOR OTHERS. • OPTIMAL SUBSTRUCTURE: REMAINING PROBLEM IS SMALLER VERSION OF SAME TASK. • SO GREEDY GIVES THE MAXIMUM NUMBER OF ACTIVITIES.
• 6.
  ELEMENTS OF GREEDYSTRATEGY • 1. GREEDY-CHOICE PROPERTY: MAKING ONE LOCAL CHOICE IS ALWAYS SAFE. • 2. OPTIMAL SUBSTRUCTURE: REMAINING PROBLEM IS SIMILAR TO ORIGINAL. • 3. ITERATIVE APPROACH: KEEP APPLYING LOCAL BEST CHOICE UNTIL DONE.
• 7.
  HUFFMAN CODING • AGREEDY METHOD USED FOR DATA COMPRESSION. • ASSIGN SHORTER CODES TO MORE FREQUENT CHARACTERS. • ASSIGN LONGER CODES TO LESS FREQUENT CHARACTERS. • ENSURES MINIMUM TOTAL STORAGE LENGTH OF ENCODED DATA.
• 8.
  HUFFMAN CODING EXAMPLE •CHARACTERS: A(5), B(9), C(12), D(13), E(16), F(45). • COMBINE LOWEST FREQUENCIES BUILD BINARY TREE. → • RESULT: F GETS SHORTEST CODE, A GETS LONGEST. • USED IN ZIP, JPEG, MP3 COMPRESSION.
• 9.
  MATROIDS AND GREEDYMETHODS • MATROID = MATHEMATICAL STRUCTURE GENERALIZING INDEPENDENCE. • HELPS IN PROVING GREEDY CORRECTNESS. • EXAMPLE: INDEPENDENT SETS IN VECTOR SPACES, SPANNING TREES IN GRAPHS. • IF A PROBLEM FITS MATROID STRUCTURE, GREEDY ALGORITHM ALWAYS WORKS.
• 10.
  TASK SCHEDULING PROBLEM •WE HAVE MULTIPLE TASKS WITH DEADLINES AND PROFITS. • GOAL: MAXIMIZE PROFIT BY FINISHING TASKS BEFORE DEADLINES. • GREEDY METHOD: CHOOSE MOST PROFITABLE TASKS FIRST. • MATROID THEORY PROVES GREEDY ALWAYS GIVES THE BEST SOLUTION HERE.
• 11.
  ADVANTAGES & LIMITATIONS •ADVANTAGES: • - SIMPLE AND EASY TO IMPLEMENT. • - OFTEN VERY FAST (O(N LOG N)). • - WORKS WELL FOR MANY OPTIMIZATION PROBLEMS. • LIMITATIONS: • - DOES NOT ALWAYS GUARANTEE GLOBAL OPTIMUM. • - ONLY WORKS IF GREEDY-CHOICE PROPERTY HOLDS.