Algorithm design. algorithms, instances, correctness efficiency, order of growth, tractability a paradigm: exhaustive search

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1 today algorithms, instances, correctness eiciency, order o growth, tractability a paradigm: exhaustive search pseudocodes design o algorithms (always!) chapter 2, section 1 and 2 rom Jon Kleinberg, Eva Tardos Algorithm design any edition, any printing

2 today algorithms, instances, correctness eiciency, order o growth, tractability a paradigm: exhaustive search pseudocodes design o algorithms (always!) chapter 2 (3) and most o chapter 1 (2) rom Thomas. H. Cormen, Charles E. Leiserson, Ronald L. Rivest Introduction to algorithms any edition, any printing green chapter numbers or the second edition!

3 some logistics sessions: two hours o lectures on average per week compensate in block C one hour o instruction on average per week but some skipping in block C one extra session o instruction beore the exam sheets o the week on: exam: written exam in january: problem rom electrical engineering statements about complexity o the problem algorithm or a relaxed problem in pseudocode analysis o algorithm at home: implementation o algorithm deense o deviations rom written exam demonstration on given example

4 sea o gates

5 a nand gate

6 a simple example ab + c + d + (+g) (e+h) a g a g c d c d b e h b e h a b c d e g h

7 a simple example a b c d e g h

8 an assignment problem given: a symbolic transistor-network ( T, N, I ) a sequence o successive gate sites S T = S a b c d e g h required an assignment T S, such that ( T, N, I ) can be completed with metal strips

9 a irst attempt generate all assignments o m transistors to m sites check or each assignment whether transistors assigned to neighbors are incident to the same node i so, complete the network with metal strips is this "algorithm" correct? an algorithm is correct i, or every instantiation o the problem, the algorithm stops ater a inite number o steps, and the output is a correct answer.

10 a small example consider an example network 1 a c 2 3 b with a possible assignment check : a b c a has a contact in common with b b has a contact in common with c conclusion : the assignment is consistent! the irst attempt does not lead to a correct algorithm however, 2 2 a b c

11 second attempt generate all assignments o m oriented transistors to m sites check or each assignment whether transistors assigned to neighbors are incident to the same node with the right contact i so, complete the network with metal strips is this "algorithm" correct? YES, but many assignments are going to be generated: m! 2 m or is the algorithm suiciently eicient? many assignments do not really have a chance can we avoid most "pointless" (partial) assignments

12 second attempt generate all assignments o m oriented transistors to m sites check or each assignment whether transistors assigned to neighbors are incident to the same node with the right contact i so, complete the network with metal strips to conclude ineiciency we need an lowerbound on runtime is this "algorithm" correct? to conclude eiciency we YES, need but an upperbound on runtime many assignments are going to be generated: m! 2 m or is the algorithm suiciently eicient? many assignments do not really have a chance can we avoid most "pointless" (partial) assignments

13 asymptotic bounds ( n) c g ( n) n > no, ( n) above c g ( n) ( c > 0 ) or always stays? n O ( n) or n > n, always stays o below c g ( n) ( c > 0 )? c g ( n) ( n) an algorithm is eicient i the number o steps to inish grows at most polynomially with the input size on a random access machine

14 asymptotic tight bounds? ( ) cgn 2 c 1 g ( n) ( n) or n > n ( n) stays in between, always c1 g ( n) and c2 g( n) ( c,c positive constants) 1 2 o an algorithm is eicient i the number o steps to inish grows at most polynomially with the input size on a random access machine

15 second attempt generate all assignments o m oriented transistors to m sites check or each assignment whether transistors assigned to neighbors are incident to the same node with the right contact i so, complete the network with metal strips is this "algorithm" correct? YES, but many assignments are going to be generated: m! 2 m or is the algorithm suiciently eicient? many assignments do not really have a chance can we avoid most "pointless" (partial assignments)

16 systematic coniguration generation required: a inite sequence o objects ( a1, a2, a3,..... rom E1 E2 E3.... satisying certain constraints ) example: elementary objects : oriented transistors constraints : common nodes, no duplications a partial sequence a, a,.....,a ) a candidate set k S k S k ( 1 2 k 1 E k a i a, has not been tried and is consistent k E k

17 pseudo code block structure by indentation control structures: iteration (while, or, repeat) conditional (i... then... else) comments: ater assignment: also multiple assignments variables : local to the given procedure unless explicitely indicated arrays : - elements accessed by index between square brackets - attributes are named with the object in square brackets parameters are passed by value

18 the backtracking paradigm S1 E 1 k 1 while k > 0 while k return the list! do Sk do a k element in S k S k S k \ {a k } i ( a1, a2,.....,ak ) is a solution then k k + 1 determine k 1 put on the list Sk

19 exhaustive assignment generation b a c d e a a 12 mogelijkheden d c e b e b c c d e b e b c b c d e e d e e c d d c c

20 reormulation critique o exhaustive assignment generation a graph can have exponentially many paths most o the work is done when there is no solution input: a graph with the nodes as vertices, the transistors as edges output: a path in that graph in which every edge occurs exactly once attempt nr 4: select a node as a starting point traverse as many edges as possible to orm a maximal path i all edges are in the path, report the solution

21 generation o a maximal path MAXPAD ( V, E, v ) P U w E v while I(w) U do e U w P P e return P element van U \ {e} I(w) U het andere element van e V: vertices o the (sub)graph E: edges o the (sub)graph v: start vertex I(w): edges incident to w P: (partial) path U: edges not yet in path P

22 invariants or some special cases all vertices have even degree : any "MAXPAD" ends in the start vertex removal o all edges o "MAXPAD" rom the graph leaves a graph with only even-degree vertices exactly two vertices have odd degree : any "MAXPAD" starting at an odd-degree vertex ends in the other odd-degree vertex removal o all edges o that "MAXPAD" rom the graph leaves a graph with only even-degree vertices

23 inserting maximal paths EXTEND ( V, E, v ) P MAXPAD ( V, E, v ) i P then E w P E \ P P P EXTEND ( V, E \ P, w ) return P

24 köningsbergen

25 evaluation correct: i at most two vertices have odd degree, then "EXTEND" produces the required path i more than two vertices have odd degree, then no such path can exist eicient: create a doubly-linked list or the irst P; perorm the i -clause or every vertex on P, progressing in spite o a changing P constant amount o work per edge O (m) solution: what i the transistor network leads to a graph with more than two odd-degree vertices?

26 tricks and treats what i the network graph has no euler path? or example: a ( b+c ) + ( d+e ) a d e b c logically equivalent with a ( b+c ) + ( d+e ) a b c d e