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Lec2 Algorth
1.
Introduction to Algorithms
6.046J/18.401J LECTURE 2 Asymptotic Notation âą O-, âŠ-, and Î-notation Recurrences âą Substitution method âą Iterating the recurrence âą Recursion tree âą Master method Prof. Erik Demaine September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.1
2.
Asymptotic notation O-notation
(upper bounds): We write f(n) = O(g(n)) if there We write f(n) = O(g(n)) if there exist constants c > 0, n00 > 0 such exist constants c > 0, n > 0 such that 0 †f(n) †cg(n) for all n ℠n00.. that 0 †f(n) †cg(n) for all n ℠n September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.2
3.
Asymptotic notation O-notation
(upper bounds): We write f(n) = O(g(n)) if there We write f(n) = O(g(n)) if there exist constants c > 0, n00 > 0 such exist constants c > 0, n > 0 such that 0 †f(n) †cg(n) for all n ℠n00.. that 0 †f(n) †cg(n) for all n ℠n EXAMPLE: 2n2 = O(n3) (c = 1, n0 = 2) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.3
4.
Asymptotic notation O-notation
(upper bounds): We write f(n) = O(g(n)) if there We write f(n) = O(g(n)) if there exist constants c > 0, n00 > 0 such exist constants c > 0, n > 0 such that 0 †f(n) †cg(n) for all n ℠n00.. that 0 †f(n) †cg(n) for all n ℠n EXAMPLE: 2n2 = O(n3) (c = 1, n0 = 2) functions, not values September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.4
5.
Asymptotic notation O-notation
(upper bounds): We write f(n) = O(g(n)) if there We write f(n) = O(g(n)) if there exist constants c > 0, n00 > 0 such exist constants c > 0, n > 0 such that 0 †f(n) †cg(n) for all n â„ n00.. that 0 †f(n) †cg(n) for all n â„ n EXAMPLE: 2n2 = O(n3) (c = 1, n0 = 2) funny, âone-wayâ functions, equality not values September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.5
6.
Set definition of
O-notation O(g(n)) = { f(n) :: there exist constants O(g(n)) = { f(n) there exist constants c > 0, n00 > 0 such c > 0, n > 0 such that 0 †f(n) †cg(n) that 0 †f(n) †cg(n) for all n ℠n00 } for all n ℠n } September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.6
7.
Set definition of
O-notation O(g(n)) = { f(n) :: there exist constants O(g(n)) = { f(n) there exist constants c > 0, n00 > 0 such c > 0, n > 0 such that 0 †f(n) †cg(n) that 0 †f(n) †cg(n) for all n â„ n00 } for all n â„ n } EXAMPLE: 2n2 â O(n3) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.7
8.
Set definition of
O-notation O(g(n)) = { f(n) :: there exist constants O(g(n)) = { f(n) there exist constants c > 0, n00 > 0 such c > 0, n > 0 such that 0 †f(n) †cg(n) that 0 †f(n) †cg(n) for all n â„ n00 } for all n â„ n } EXAMPLE: 2n2 â O(n3) (Logicians: λn.2n2 â O(λn.n3), but itâs convenient to be sloppy, as long as we understand whatâs really going on.) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.8
9.
Macro substitution Convention: A
set in a formula represents an anonymous function in the set. September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.9
10.
Macro substitution Convention: A
set in a formula represents an anonymous function in the set. EXAMPLE: f(n) = n3 + O(n2) means f(n) = n3 + h(n) for some h(n) â O(n2) . September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.10
11.
Macro substitution Convention: A
set in a formula represents an anonymous function in the set. EXAMPLE: n2 + O(n) = O(n2) means for any f(n) â O(n): n2 + f(n) = h(n) for some h(n) â O(n2) . September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.11
12.
âŠ-notation (lower bounds) O-notation
is an upper-bound notation. It makes no sense to say f(n) is at least O(n2). September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.12
13.
âŠ-notation (lower bounds) O-notation
is an upper-bound notation. It makes no sense to say f(n) is at least O(n2). âŠ(g(n)) = { f(n) :: there exist constants âŠ(g(n)) = { f(n) there exist constants c > 0, n00 > 0 such c > 0, n > 0 such that 0 †cg(n) †f(n) that 0 †cg(n) †f(n) for all n â„ n00 } for all n â„ n } September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.13
14.
âŠ-notation (lower bounds) O-notation
is an upper-bound notation. It makes no sense to say f(n) is at least O(n2). âŠ(g(n)) = { f(n) :: there exist constants âŠ(g(n)) = { f(n) there exist constants c > 0, n00 > 0 such c > 0, n > 0 such that 0 †cg(n) †f(n) that 0 †cg(n) †f(n) for all n â„ n00 } for all n â„ n } EXAMPLE: n = âŠ(lg n) (c = 1, n0 = 16) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.14
15.
Î-notation (tight bounds)
Î(g(n)) = O(g(n)) â© âŠ(g(n)) Î(g(n)) = O(g(n)) â© âŠ(g(n)) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.15
16.
Î-notation (tight bounds)
Î(g(n)) = O(g(n)) â© âŠ(g(n)) Î(g(n)) = O(g(n)) â© âŠ(g(n)) EXAMPLE: 1 2 n â 2 n = Î( n ) 2 2 September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.16
17.
Îż-notation and Ï-notation O-notation
and âŠ-notation are like †and â„. o-notation and Ï-notation are like < and >. Îż(g(n)) = { f(n) :: for any constant c > 0, Îż(g(n)) = { f(n) for any constant c > 0, there is a constant n00 > 0 there is a constant n > 0 such that 0 †f(n) < cg(n) such that 0 †f(n) < cg(n) for all n â„ n00 } for all n â„ n } EXAMPLE: 2n2 = o(n3) (n0 = 2/c) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.17
18.
Îż-notation and Ï-notation O-notation
and âŠ-notation are like †and â„. o-notation and Ï-notation are like < and >. Ï(g(n)) = { f(n) :: for any constant c > 0, Ï(g(n)) = { f(n) for any constant c > 0, there is a constant n00 > 0 there is a constant n > 0 such that 0 †cg(n) < f(n) such that 0 †cg(n) < f(n) for all n â„ n00 } for all n â„ n } EXAMPLE: n = Ï (lg n) (n0 = 1+1/c) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.18
19.
Solving recurrences
⹠The analysis of merge sort from Lecture 1 required us to solve a recurrence. ⹠Recurrences are like solving integrals, differential equations, etc. o Learn a few tricks. ⹠Lecture 3: Applications of recurrences to divide-and-conquer algorithms. September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.19
20.
Substitution method
The most general method: 1. Guess the form of the solution. 2. Verify by induction. 3. Solve for constants. September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.20
21.
Substitution method
The most general method: 1. Guess the form of the solution. 2. Verify by induction. 3. Solve for constants. EXAMPLE: T(n) = 4T(n/2) + n âą [Assume that T(1) = Î(1).] âą Guess O(n3) . (Prove O and ⊠separately.) âą Assume that T(k) †ck3 for k < n . âą Prove T(n) †cn3 by induction. September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.21
22.
Example of substitution T
(n) = 4T (n / 2) + n †4c ( n / 2 ) 3 + n = ( c / 2) n 3 + n = cn3 â ((c / 2)n3 â n) desired â residual †cn3 desired whenever (c/2)n3 â n â„ 0, for example, if c â„ 2 and n â„ 1. residual September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.22
23.
Example (continued) âą
We must also handle the initial conditions, that is, ground the induction with base cases. âą Base: T(n) = Î(1) for all n < n0, where n0 is a suitable constant. âą For 1 †n < n0, we have âÎ(1)â †cn3, if we pick c big enough. September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.23
24.
Example (continued) âą
We must also handle the initial conditions, that is, ground the induction with base cases. âą Base: T(n) = Î(1) for all n < n0, where n0 is a suitable constant. âą For 1 †n < n0, we have âÎ(1)â †cn3, if we pick c big enough. This bound is not tight! September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.24
25.
A tighter upper
bound? We shall prove that T(n) = O(n2). September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.25
26.
A tighter upper
bound? We shall prove that T(n) = O(n2). Assume that T(k) †ck2 for k < n: T (n) = 4T (n / 2) + n †4 c ( n / 2) 2 + n = cn 2 + n = O(n 2 ) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.26
27.
A tighter upper
bound? We shall prove that T(n) = O(n2). Assume that T(k) †ck2 for k < n: T (n) = 4T (n / 2) + n †4c ( n / 2) 2 + n = cn 2 + n = O(n 2 ) Wrong! We must prove the I.H. September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.27
28.
A tighter upper
bound? We shall prove that T(n) = O(n2). Assume that T(k) †ck2 for k < n: T (n) = 4T (n / 2) + n †4c ( n / 2) 2 + n = cn 2 + n = O(n 2 ) Wrong! We must prove the I.H. = cn 2 â (â n) [ desired â residual ] †cn 2 for no choice of c > 0. Lose! September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.28
29.
A tighter upper
bound! IDEA: Strengthen the inductive hypothesis. âą Subtract a low-order term. Inductive hypothesis: T(k) †c1k2 â c2k for k < n. September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.29
30.
A tighter upper
bound! IDEA: Strengthen the inductive hypothesis. âą Subtract a low-order term. Inductive hypothesis: T(k) †c1k2 â c2k for k < n. T(n) = 4T(n/2) + n = 4(c1(n/2)2 â c2(n/2)) + n = c1n2 â 2c2n + n = c1n2 â c2n â (c2n â n) †c1n2 â c2n if c2 â„ 1. September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.30
31.
A tighter upper
bound! IDEA: Strengthen the inductive hypothesis. âą Subtract a low-order term. Inductive hypothesis: T(k) †c1k2 â c2k for k < n. T(n) = 4T(n/2) + n = 4(c1(n/2)2 â c2(n/2)) + n = c1n2 â 2c2n + n = c1n2 â c2n â (c2n â n) †c1n2 â c2n if c2 â„ 1. Pick c1 big enough to handle the initial conditions. September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.31
32.
Recursion-tree method âą A
recursion tree models the costs (time) of a recursive execution of an algorithm. âą The recursion-tree method can be unreliable, just like any method that uses ellipses (âŠ). âą The recursion-tree method promotes intuition, however. âą The recursion tree method is good for generating guesses for the substitution method. September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.32
33.
Example of recursion
tree Solve T(n) = T(n/4) + T(n/2) + n2: September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.33
34.
Example of recursion
tree Solve T(n) = T(n/4) + T(n/2) + n2: T(n) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.34
35.
Example of recursion
tree Solve T(n) = T(n/4) + T(n/2) + n2: n2 T(n/4) T(n/2) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.35
36.
Example of recursion
tree Solve T(n) = T(n/4) + T(n/2) + n2: n2 (n/4)2 (n/2)2 T(n/16) T(n/8) T(n/8) T(n/4) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.36
37.
Example of recursion
tree Solve T(n) = T(n/4) + T(n/2) + n2: n2 (n/4)2 (n/2)2 (n/16)2 (n/8)2 (n/8)2 (n/4)2 ⊠Î(1) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.37
38.
Example of recursion
tree Solve T(n) = T(n/4) + T(n/2) + n2: n2 n2 (n/4)2 (n/2)2 (n/16)2 (n/8)2 (n/8)2 (n/4)2 ⊠Î(1) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.38
39.
Example of recursion
tree Solve T(n) = T(n/4) + T(n/2) + n2: n2 n2 5 n2 (n/4)2 (n/2)2 16 (n/16)2 (n/8)2 (n/8)2 (n/4)2 ⊠Î(1) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.39
40.
Example of recursion
tree Solve T(n) = T(n/4) + T(n/2) + n2: n2 n2 5 n2 (n/4)2 (n/2)2 16 25 n 2 (n/16)2 (n/8)2 (n/8)2 (n/4)2 256 ⊠⊠Î(1) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.40
41.
Example of recursion
tree Solve T(n) = T(n/4) + T(n/2) + n2: n2 n2 5 n2 (n/4)2 (n/2)2 16 25 n 2 (n/16)2 (n/8)2 (n/8)2 (n/4)2 256 ⊠⊠Î(1) Total = n 2 ( 5 + 5 2 1 + 16 16 ( ) +( ) +L 5 3 16 ) = Î(n2) geometric series September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.41
42.
The master method The
master method applies to recurrences of the form T(n) = a T(n/b) + f (n) , where a ℠1, b > 1, and f is asymptotically positive. September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.42
43.
Three common cases Compare
f (n) with nlogba: 1. f (n) = O(nlogba â Δ) for some constant Δ > 0. âą f (n) grows polynomially slower than nlogba (by an nΔ factor). Solution: T(n) = Î(nlogba) . September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.43
44.
Three common cases Compare
f (n) with nlogba: 1. f (n) = O(nlogba â Δ) for some constant Δ > 0. âą f (n) grows polynomially slower than nlogba (by an nΔ factor). Solution: T(n) = Î(nlogba) . 2. f (n) = Î(nlogba lgkn) for some constant k â„ 0. âą f (n) and nlogba grow at similar rates. Solution: T(n) = Î(nlogba lgk+1n) . September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.44
45.
Three common cases
(cont.) Compare f (n) with nlogba: 3. f (n) = âŠ(nlogba + Δ) for some constant Δ > 0. âą f (n) grows polynomially faster than nlogba (by an nΔ factor), and f (n) satisfies the regularity condition that a f (n/b) †c f (n) for some constant c < 1. Solution: T(n) = Î( f (n)) . September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.45
46.
Examples EX. T(n) =
4T(n/2) + n a = 4, b = 2 â nlogba = n2; f (n) = n. CASE 1: f (n) = O(n2 â Δ) for Δ = 1. ⎠T(n) = Î(n2). September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.46
47.
Examples EX. T(n) =
4T(n/2) + n a = 4, b = 2 â nlogba = n2; f (n) = n. CASE 1: f (n) = O(n2 â Δ) for Δ = 1. ⎠T(n) = Î(n2). EX. T(n) = 4T(n/2) + n2 a = 4, b = 2 â nlogba = n2; f (n) = n2. CASE 2: f (n) = Î(n2lg0n), that is, k = 0. ⎠T(n) = Î(n2lg n). September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.47
48.
Examples EX. T(n) =
4T(n/2) + n3 a = 4, b = 2 â nlogba = n2; f (n) = n3. CASE 3: f (n) = âŠ(n2 + Δ) for Δ = 1 and 4(n/2)3 †cn3 (reg. cond.) for c = 1/2. ⎠T(n) = Î(n3). September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.48
49.
Examples EX. T(n) =
4T(n/2) + n3 a = 4, b = 2 â nlogba = n2; f (n) = n3. CASE 3: f (n) = âŠ(n2 + Δ) for Δ = 1 and 4(n/2)3 †cn3 (reg. cond.) for c = 1/2. ⎠T(n) = Î(n3). EX. T(n) = 4T(n/2) + n2/lg n a = 4, b = 2 â nlogba = n2; f (n) = n2/lg n. Master method does not apply. In particular, for every constant Δ > 0, we have nΔ = Ï(lg n). September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.49
50.
Idea of master
theorem Recursion tree: a f (n) f (n/b) f (n/b) ⊠f (n/b) a f (n/b2) f (n/b2) ⊠f (n/b2) ⊠΀ (1) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.50
51.
Idea of master
theorem Recursion tree: f (n) f (n) a f (n/b) f (n/b) ⊠f (n/b) a f (n/b) a f (n/b2) f (n/b2) ⊠f (n/b2) a2 f (n/b2) ⊠⊠΀ (1) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.51
52.
Idea of master
theorem Recursion tree: f (n) f (n) a f (n/b) f (n/b) ⊠f (n/b) a f (n/b) h = logbn a f (n/b2) f (n/b2) ⊠f (n/b2) a2 f (n/b2) ⊠⊠΀ (1) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.52
53.
Idea of master
theorem Recursion tree: f (n) f (n) a f (n/b) f (n/b) ⊠f (n/b) a f (n/b) h = logbn a f (n/b2) f (n/b2) ⊠f (n/b2) a2 f (n/b2) #leaves = ah ⊠⊠= alogbn ΀ (1) nlogba΀ (1) = nlogba September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.53
54.
Idea of master
theorem Recursion tree: f (n) f (n) a f (n/b) f (n/b) ⊠f (n/b) a f (n/b) h = logbn a f (n/b2) f (n/b2) ⊠f (n/b2) a2 f (n/b2) ⊠CASE 1: The weight increases ⊠CASE 1: The weight increases geometrically from the root to the geometrically from the root to the ΀ (1) leaves. The leaves hold aaconstant leaves. The leaves hold constant nlogba΀ (1) fraction of the total weight. fraction of the total weight. Î(nlogba) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.54
55.
Idea of master
theorem Recursion tree: f (n) f (n) a f (n/b) f (n/b) ⊠f (n/b) a f (n/b) h = logbn a f (n/b2) f (n/b2) ⊠f (n/b2) a2 f (n/b2) ⊠⊠CASE 2: (k = 0) The weight CASE 2: (k = 0) The weight ΀ (1) is approximately the same on is approximately the same on nlogba΀ (1) each of the logbbn levels. each of the log n levels. Î(nlogbalg n) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.55
56.
Idea of master
theorem Recursion tree: f (n) f (n) a f (n/b) f (n/b) ⊠f (n/b) a f (n/b) h = logbn a f (n/b2) f (n/b2) ⊠f (n/b2) a2 f (n/b2) ⊠CASE 3: The weight decreases ⊠CASE 3: The weight decreases geometrically from the root to the geometrically from the root to the ΀ (1) leaves. The root holds aaconstant leaves. The root holds constant nlogba΀ (1) fraction of the total weight. fraction of the total weight. Î( f (n)) September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.56
57.
Appendix: geometric series
2 n 1 â x n +1 1+ x + x +L+ x = for x â 1 1â x 21 1+ x + x +L = for |x| < 1 1â x Return to last slide viewed. September 12, 2005 Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson L2.57
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