Algorithms, Part II Final Exam Part II

If there are two edge-disjoint paths connecting vertices u and v, then there is a simple cycle containing both u and v.

If you delete two edges from a graph G, its number of strong components can increase by at most 2.

A graph G that has a unique simple path between every pair of vertices has exactly V-1 edges.

Given a graph G and a vertex s, G is connected if and only if every vertex is connected to s.

Removing any edge from a connected graph G breaks the graph into two connected components.

If a vertex has no edges pointing from it, it will be first in any postorder

The postorder of a digraph G is the reverse of its preorder.

If there is a unique vertex reachable from every other vertex in a digraph, it will be first in some postorder.

The reverse postorder of a digraph's reverse is the same as the postorder of the digraph.

If a digraph has a vertex with no edges pointing from it, then some vertex with no edges pointing from it will be first in any postorder.

Let G be a connected edge-weighted graph and let T be a MST of G. Then, the cheapest spanning tree of G besides T can be obtained by deleting the most expensive edge in T and computing the MST of the resulting graph.

Let G be a connected edge-weighted graph and let T be an MST of G. If you decrease the weight of some edge in T, then T will still be a MST.

Let G be a connected edge-weighted graph with distinct edge weights and no parallel edges. Then, the MST must contain the third cheapest edge.

Let T and T' be two MSTs of some edge-weighted graph and let a[] and b[] be the sorted array of edge weights. Then a[i] = b[i] for each i.

Let G be a connected edge-weighted graph with distinct edge weights and let (A, B) be a cut with at least 3 crossing edges. Then, the MST can contain the smallest and third smallest crossing edges without containing the second smallest crossing edge.

If you run the Bellman-Ford algorithm on an edge-weighted digraph with positive edge weights and the shortest path from s to t has k edges, then Bellman-Ford will require at least k passes.

Bellman-Ford finds the shortest simple path from s to every other vertex, even if the edge weights are arbritary positive or negative integers.

Bellman-Ford finds the shortest simple path from s to every other vertex, even if the edge weights are positive or negative integers, provided there are no negative cycles.

Let G be a digraph with positive edge weights. During Bellman-Ford, if you trace back the edgeTo[] entries starting from v back to s, the path has distance equal to distTo[v].

Let G be a digraph with positive edge weights. Suppose that you increase the length of an edge by x. Then, the length of the shortest path from s to t can increase by more than x.

If all edge capacities are increased by an additive constant, then any mincut in the original network is a mincut in the modified network.

There exists a maxflow for which there is no directed cycle on which every edge carries positive flow.

Given a mincut (S, T) if the capacity of one edge is decreased by x, the value of the maxflow will decrease by exactly x.

If all edge capacities are integral, then any maxflow has integer flow on each edge.

If the capacity of some cut (S, T) equals the value of some flow f, then f is a maxflow.

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The column on the left contains the original input of 24 strings to be sorted;
the column on the right contains the strings in sorted order; the other 7 columns contain the
contents at some intermediate step during one of the 3 radix sorting algorithms listed below.

    moth   lamb   boar   puma   ibex   toad   bass   boar   bass   
    puma   hare   bass   tuna   boar   clam   boar   dove   boar   
    ibex   carp   carp   lamb   dove   boar   carp   bass   carp   
    boar   bass   clam   toad   kiwi   gnat   clam   carp   clam   
    dove   ibex   dove   dove   carp   wren   dove   erne   dove   
    kiwi   mink   erne   sole   erne   ibex   erne   clam   erne   
    carp   lion   gnat   hare   lion   sole   gnat   hare   gnat   
    tuna   kiwi   hare   mole   lamb   mole   hare   gnat   hare   
    sole   clam   ibex   erne   gnat   mule   ibex   ibex   ibex   
    lion   slug   kiwi   mule   loon   puma   kiwi   loon   kiwi   
    lamb   gnat   lion   slug   hare   lamb   lion   lamb   lamb   
    wren   toad   lamb   moth   clam   tuna   lamb   lion   lion   
    gnat   boar   loon   kiwi   bass   erne   loon   kiwi   loon   
    loon   sole   moth   mink   moth   mink   moth   moth   mink   
    hare   mole   mole   clam   mink   lion   mole   mink   mole   
    clam   loon   mink   lion   mole   loon   mink   mole   moth   
    bass   moth   mule   wren   mule   hare   mule   mule   mule   
    oryx   dove   oryx   loon   slug   carp   oryx   slug   oryx   
    slug   wren   puma   carp   oryx   bass   puma   oryx   puma   
    mole   erne   sole   boar   toad   moth   sole   toad   slug   
    toad   oryx   slug   bass   wren   slug   slug   wren   sole   
    erne   mule   tuna   gnat   sole   dove   tuna   sole   toad   
    mink   puma   toad   ibex   tuna   kiwi   toad   tuna   tuna   
    mule   tuna   wren   oryx   puma   oryx   wren   puma   wren   
    ----   ----   ----   ----   ----   ----   ----   ----   ----   
     0      ?      ?      ?      ?      ?      ?      ?      4     

Match up each column with the corresponding sorting algorithm from the given list:

    0. Original input
    1. LSD radix sort
    2. MSD radix sort
    3. 3-way radix quicksort (no shuffle)
    4. Sorted

You may use an algorithm more than once. Your answer should be a sequence of 9 integers between
0 and 4 (starting with 0 and ending with 4) and with each integer separated by a single space.

Hint: think about algorithm invariants. Do not trace code.

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What is the Burrows-Wheeler inverse transform of the following?

    5  B D A C D D A A 

Your answer should be a sequence of 8 characters, with a single space
separating each character.

Computing the remainder when dividing one N-bit integer into an N-bit integer.

Adding two N-bit integers.

Computing the remainder when dividing an N-bit integer by 17.

Squaring an N-bit integer.

Factoring an N-bit integer.

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Consider the following linear programming simplex tableaux with 6 equations and 8 variables:

    maximize Z
                -    3 x1                                                         +  3/4 x7    -  Z    = -156
    ---------------------------------------------------------------------------------------------------------
                +    4 x1                                   +    1 x5             +    1 x7            =   18
     +    1 x0  -    1 x1                                                         -  6/5 x7            =   24
                +  3/2 x1                        +    1 x4                        +  3/5 x7            =   42
                +  5/4 x1  +    1 x2                                              +  1/2 x7            =   48
                +    2 x1                                              +    1 x6  -  5/2 x7            =   30
                +  5/2 x1             +    1 x3                                                        =   48
            x0  ,      x1  ,      x2  ,      x3  ,      x4  ,      x5  ,      x6  ,      x7           >=    0

What is the value of variable x5 in the current basic feasible solution?
Specify your answer with two digits after the decimal place.

linear time

constant time

exponential space

exponential time

polynomial time

Given an undirected graph and two vertices s and t, find a path from s to t in E + V time.

Given an undirected graph, design an E + V algorithm to find a spanning tree.

Given an undirected graph, find an edge whose removal increases the number of connected components in (E+V) E time.

Given an digraph, design an E V algorithm to find the shortest directed cycle.

Given a digraph with positive edge weights and two vertices s and t, design an E + V algorithm to find a max st-flow.

Given an array of N strings, sort them in time linear in N in the worst case.

Determine whether a pattern string of length M is a substring of a text string of length N using a constant amount of extra memory.

Compute the suffix array of a string of length N in time proportional to N log N in the worst case.

Determine whether an M-character regular expression matches an N-character text in time polynomial in M and N.

Construct a DFA corresponding to an M-character regular expression.

Question 1

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Which of the following statements about connectivity in an undirected graph are true? Check all that apply.

If there are two edge-disjoint paths connecting vertices u and v, then there is a simple cycle containing both u and v.

If you delete two edges from a graph, its number of connected components can increase by at most 2.

A graph that has a unique simple path between every pair of vertices has exactly V-1 edges.

Given a graph and a vertex s, the graph has one connected component if and only if every vertex is connected to s.

Removing any edge from a connected graph breaks the graph into two connected components.

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Which of the following statements about depth-first search in a digraph are guaranteed to be true? Check all that apply.

If a vertex has outdegree 0, then it will be first in any postorder.

The postorder of a digraph is the reverse of its preorder.

If there is a vertex reachable from every other vertex, it will be first in some postorder.

The reverse postorder of G^R is equal to the postorder of G.

If there is a vertex with outdegree 0, then some vertex with outdegree 0 will be first in any postorder.

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Which of the following statements about minimum spanning trees (MSTs) are guaranteed to be true? Check all that apply.

Let G be a connected edge-weighted graph and let T be a MST of G. Then, the cheapest spanning tree of G besides T can be obtained by deleting the most expensive edge in T and computing the MST of the resulting graph.

Let G be a connected edge-weighted graph and let T be an MST of G. If you decrease the weight of some edge in T, then T will still be a MST.

Let G be a connected edge-weighted graph with distinct edge weights and no parallel edges. Then, the MST must contain the third cheapest edge.

Let T and T' be two MSTs of some edge-weighted graph and let a[] and b[] be the sorted array of edge weights for T and T', respectively. Then a[i] = b[i] for each i.

Let G be a connected edge-weighted graph with distinct edge weights and let (A, B) be a cut with at least 3 crossing edges. Then, the MST can contain the smallest and third smallest crossing edges without containing the second smallest crossing edge.

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Which of the following statements about shortest paths are true? Check all that apply.

If you run the Bellman-Ford algorithm on an edge-weighted digraph with positive edge weights and the shortest path from s to t has k edges, then Bellman-Ford will require at least k passes.

Bellman-Ford finds the shortest simple path from s to every other vertex, even if the edge weights are arbritary positive or negative integers.

Bellman-Ford finds the shortest simple path from s to every other vertex, even if the edge weights are positive or negative integers, provided there are no negative cycles.

Let G be a digraph with positive edge weights. During Bellman-Ford, if you trace back the edgeTo[] entries starting from v back to s, the path has distance equal to distTo[v].

Let G be a digraph with positive edge weights. Suppose that you increase the length of an edge by x. Then, the length of the shortest path from s to t can increase by more than x.

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Which of the following statements about maxflow and mincut are guaranteeed to be true? Check all that apply.

If all edge capacities are increased by an additive constant, then any mincut in the original network is a mincut in the modified network.

There exists a maxflow for which there is no directed cycle on which every edge carries positive flow.

Given a mincut (A, B), if the capacity of one edge from A to B is decreased by x, then the value of the maxflow will decrease by exactly x.

If all edge capacities are integral, then any maxflow has integer flow on each edge.

If the capacity of some cut (A, B) equals the value of some flow f, then f is a maxflow.

(seed = 725630)
The column on the left contains the original input of 24 strings to be sorted;
the column on the right contains the strings in sorted order; the other 7 columns contain the
contents at some intermediate step during one of the 3 radix sorting algorithms listed below.

    moth   lamb   boar   puma   ibex   toad   bass   boar   bass   
    puma   hare   bass   tuna   boar   clam   boar   dove   boar   
    ibex   carp   carp   lamb   dove   boar   carp   bass   carp   
    boar   bass   clam   toad   kiwi   gnat   clam   carp   clam   
    dove   ibex   dove   dove   carp   wren   dove   erne   dove   
    kiwi   mink   erne   sole   erne   ibex   erne   clam   erne   
    carp   lion   gnat   hare   lion   sole   gnat   hare   gnat   
    tuna   kiwi   hare   mole   lamb   mole   hare   gnat   hare   
    sole   clam   ibex   erne   gnat   mule   ibex   ibex   ibex   
    lion   slug   kiwi   mule   loon   puma   kiwi   loon   kiwi   
    lamb   gnat   lion   slug   hare   lamb   lion   lamb   lamb   
    wren   toad   lamb   moth   clam   tuna   lamb   lion   lion   
    gnat   boar   loon   kiwi   bass   erne   loon   kiwi   loon   
    loon   sole   moth   mink   moth   mink   moth   moth   mink   
    hare   mole   mole   clam   mink   lion   mole   mink   mole   
    clam   loon   mink   lion   mole   loon   mink   mole   moth   
    bass   moth   mule   wren   mule   hare   mule   mule   mule   
    oryx   dove   oryx   loon   slug   carp   oryx   slug   oryx   
    slug   wren   puma   carp   oryx   bass   puma   oryx   puma   
    mole   erne   sole   boar   toad   moth   sole   toad   slug   
    toad   oryx   slug   bass   wren   slug   slug   wren   sole   
    erne   mule   tuna   gnat   sole   dove   tuna   sole   toad   
    mink   puma   toad   ibex   tuna   kiwi   toad   tuna   tuna   
    mule   tuna   wren   oryx   puma   oryx   wren   puma   wren   
    ----   ----   ----   ----   ----   ----   ----   ----   ----   
     0      ?      ?      ?      ?      ?      ?      ?      4     


Match up each column with the corresponding sorting algorithm from the given list:

    0. Original input
    1. LSD radix sort
    2. MSD radix sort
    3. 3-way radix quicksort (no shuffle)
    4. Sorted

You may use an algorithm more than once. Your answer should be a sequence of 9 integers between
0 and 4 (starting with 0 and ending with 4) and with each integer separated by a single space.

Hint: think about algorithm invariants. Do not trace code.

The correct answer is:0 1 2 1 3 1 2 3 4

 0: Original input
 1: LSD radix sort after 3 passes
 2: MSD radix sort after the first call to key-indexed counting
 1: LSD radix sort after 1 pass
 3: 3-way radix quicksort after the first partitioning step
 1: LSD radix sort after 2 passes
 2: MSD radix sort after the second call to key-indexed counting
 3: 3-way radix quicksort after the second partitioning step
 4: Sorted

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What is the Burrows-Wheeler inverse transform of the following?

    5  B D A C D D A A 

Your answer should be a sequence of 8 characters, with a single space
separating each character.

The correct answer is:D A D C B A A D


Below are the sorted cyclic suffixes and the next[] array:

  i  suffixes[i]   t   next[i]
 -----------------------------
  0  A ? ? ? ? ? ? B     2
  1  A ? ? ? ? ? ? D     6
  2  A ? ? ? ? ? ? A     7
  3  B ? ? ? ? ? ? C     0
  4  C ? ? ? ? ? ? D     3
* 5  D ? ? ? ? ? ? D     1
  6  D ? ? ? ? ? ? A     4
  7  D ? ? ? ? ? ? A     5

Since first = 5, the first character is D (the first character in row 5).

Since next[5] = 1, the second character is A (the first character in row 1).

Since next[next[5]] = 6, the third character is D (the first character in row 6).

Since next[next[next[5]]] = 4, the fourth character is C (the first character in row 4).

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Which problems are known to have the same asymptotic complexity as multiplying two N-bit integers? Check all that apply.

Computing the remainder when dividing one N-bit integer into an N-bit integer.

Adding two N-bit integers.

Computing the remainder when dividing an N-bit integer by 17.

Squaring an N-bit integer.

Factoring an N-bit integer.

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Consider the following linear programming simplex tableaux with 6 equations and 8 variables:

    maximize Z
                -    3 x1                                                         +  3/4 x7    -  Z    = -156
    ---------------------------------------------------------------------------------------------------------
                +    4 x1                                   +    1 x5             +    1 x7            =   18
     +    1 x0  -    1 x1                                                         -  6/5 x7            =   24
                +  3/2 x1                        +    1 x4                        +  3/5 x7            =   42
                +  5/4 x1  +    1 x2                                              +  1/2 x7            =   48
                +    2 x1                                              +    1 x6  -  5/2 x7            =   30
                +  5/2 x1             +    1 x3                                                        =   48
            x0  ,      x1  ,      x2  ,      x3  ,      x4  ,      x5  ,      x6  ,      x7           >=    0


What is the value of variable x5 in the current basic feasible solution?
Specify your answer with two digits after the decimal place.

The basis is { x5, x0, x4, x2, x6, x3 }.
The nonbasic variables are { x1, x7 }.
The basic variable x5 appears in row 0.
Its current value equals the value of the RHS entry in that row, which is 18.

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A problem is intractable if and only if it cannot be solved in

linear time

constant time

exponential space

exponential time

polynomial time

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You are applying for a job at a new software technology company. Your interviewer asks you
to identify which of the following tasks are known to be possible. Check all that apply.

Given an undirected graph and two vertices s and t, find a path from s to t in E + V time.

Given an undirected graph, design an E + V algorithm to find a spanning tree.

Given an undirected graph, find an edge whose removal increases the number of connected components in (E+V) E time.

Given an digraph, design an E V algorithm to find the shortest directed cycle.

Given a digraph with positive edge weights and two vertices s and t, design an E + V algorithm to find a max st-flow.

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You are applying for a job at a new software technology company. Your interviewer asks you
to identify which of the following tasks are known to be possible. Check all that apply.

Given an array of N strings, sort them in time linear in N in the worst case.

Determine whether a pattern string of length M is a substring of a text string of length N using a constant amount of extra memory.

Compute the suffix array of a string of length N in time proportional to N log N in the worst case.

Determine whether an M-character regular expression matches an N-character text in time polynomial in M and N.

Construct a DFA corresponding to an M-character regular expression.