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Understanding Algorithms: Problem Solving, Efficiency, and Classification

This overview provides a comprehensive introduction to algorithms, including methods for solving problems with finite steps, their efficiency, and various algorithm types. Key concepts include Big O, Θ, and Ω notation for measuring performance, and examples like Binary Search and Merge Sort to illustrate different approaches to problem-solving. The discussion on tractable versus intractable problems highlights the significance of polynomial time in algorithm classification, including the famous P vs. NP problem, which remains one of the biggest challenges in computer science.

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Understanding Algorithms: Problem Solving, Efficiency, and Classification

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  1. Algorithms Matt Schierholtz

  2. Overview • Method for solving problems with finite steps • Algorithm example: • Error • Check for problem • Solve problem • Must terminate

  3. Efficiency • Algorithms vary greatly in either memory space or time required • Sorted into worst, best and average performance • Big O, Big Θ and Big Ω notation • O(f(x)) is at most f(x)

  4. Algorithm Types • Nested loop • s = 0 • for i = 1 to n • for j = 1 to i • s = s + j(i – j + 1) • next j • next i • Find the number of steps this algorithm takes • Compare efficiency of particular algorithms

  5. Algorithms and Logarithms • Logbx = a is very useful in computer calculations • For example, • Log21024 = 10 and log21048576 = 20 • ⌊a⌋ is the number of bits used to represent a number x in binary, or ternary, or any number base ya want

  6. Binary Search • Takes less time than sequential search • Trades time for organization • Example • While (top >= bot and index = 0) • Mid = ⌊(bot + top) / 2⌋ • If a[mid] = x then index = mid • If a[mid] > x • Then top = mid - 1 • Else bot = mid + 1 • End while

  7. Merge Sort • Easier to write than Binary search • But takes more time • Think recursively • Suppose: • Efficient algorithm for arrays < k known • What if you have array < k?

  8. More Merge Sort • Sort smaller parts! • Then you merge Initial number group Get sorted Merge

  9. Tractable & Intractable Problems • Algorithms with exponential order • Ex: Tower of Hanoi • If it has 64 disks • Number of moves = 264 – 1 • For a computer, moves / second = 109 • This will take 584 years

  10. P vs. NP • Polynomial algorithms are class P • These are tractable • Even if they take a very long time • Problems not solved in polynomial time are… • Intractable • Class NP (nondeterministic polynomial) • $1,000,000, to anyone who proves P=NP • NP-Complete • If one NP problem is solvable, all of them are

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