Data Structures and Algorithms: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

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1Formulas and methods

Graph theory

Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij

Method

  1. Write down the degree of every vertex; the degrees add up to twice the number of edges.
  2. For Euler questions, check that the graph is connected and count the vertices with odd degree.
  3. For trees, use n − 1 edges and the handshake lemma.
  4. For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
  5. For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).

2Practice problems

1.Algorithms & complexityMedium

With coins of 1, 3 and 4 and an amount of 6, how many coins does the greedy "largest coin first" method use, compared to the optimum?

  1. AGreedy uses 4 coins; the optimum is 3
  2. BGreedy uses 2 coins; the optimum is 2
  3. CGreedy uses 3 coins; the optimum is 2
  4. DGreedy uses 3 coins; the optimum is 3
2.Algorithms & complexityHard

What would follow if someone found a polynomial-time algorithm for one NP-complete problem?

  1. AEvery problem in NP could be solved in polynomial time, so P = NP
  2. BThe halting problem would become decidable
  3. COnly that one problem would move into P
  4. DAll exponential-time algorithms would immediately run in linear time
3.Data structuresMedium

A trie (prefix tree) is especially efficient for:

  1. APrefix searches such as autocomplete
  2. BSorting numbers
  3. CFinding shortest paths
  4. DStoring matrices
4.Data structuresHard

In a hash table using open addressing, why is deletion often done with a "tombstone" marker?

  1. AIt makes the table sorted
  2. BTombstones speed up hashing
  3. CEmptying the slot could break the probe sequence for other keys
  4. DDeleted keys must be kept forever for security, so slots can never be reused
5.Graph theoryMedium

A connected graph has 17 vertices and 25 edges. How many edges k must be removed to leave a spanning tree?

|V| = 17|E| = 25
k =
6.Graph theoryHard
ABCDEF

How many walks of length 3 go from A to F? A walk may repeat vertices and edges. Call the number N.

M = adjacency matrixL = 3
N =

3Solutions

1.Algorithms & complexity

Answer: C · Greedy uses 3 coins; the optimum is 2

Why this is the answer

Greedy takes 4 + 1 + 1 = 3 coins, but 3 + 3 uses only 2, so greedy is not optimal for this coin system; dynamic programming is.

Answer

C · Greedy uses 3 coins; the optimum is 2
2.Algorithms & complexity

Answer: A · Every problem in NP could be solved in polynomial time, so P = NP

Why this is the answer

Since all of NP reduces to any NP-complete problem in polynomial time, a fast algorithm for one gives a fast algorithm for all.

Answer

A · Every problem in NP could be solved in polynomial time, so P = NP
3.Data structures

Answer: A · Prefix searches such as autocomplete

Why this is the answer

Tries store strings by characters along paths, so lookups and prefix queries take time proportional to the key length.

Answer

A · Prefix searches such as autocomplete
4.Data structures

Answer: C · Emptying the slot could break the probe sequence for other keys

Why this is the answer

If a deleted slot became empty, later searches might stop early and miss keys that were placed further along the probe chain.

Answer

C · Emptying the slot could break the probe sequence for other keys
5.Graph theory

Answer: k = 9

Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.

Edges of a spanning tree

17 − 1 = 16

Edges to remove

k = 25 − 16 = 9

Answer

k = 9
6.Graph theory

Answer: N = 1

Degrees: the degree of a vertex is the number of edges that meet it. Every edge has two ends, so the degrees add up to twice the number of edges: Σ deg(v) = 2|E| (the handshake lemma). The number of vertices with odd degree is therefore even. Kn has n(n − 1)/2 edges, a k-regular graph on n vertices has nk/2, and a tree on n vertices has n − 1 edges.

Adjacency matrix

A: 0 0 1 0 0 0
B: 0 0 1 1 1 0
C: 1 1 0 0 1 0
D: 0 1 0 0 1 0
E: 0 1 1 1 0 1
F: 0 0 0 0 1 0

One more step

(M³)AF = 1·0 + 1·0 + 0·0 + 0·0 + 1·1 + 0·0 = 1

Answer

N = 1

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