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

Which statement about little-o notation is correct?

  1. An² = o(n)
  2. Bn = o(n²), but n is not o(n)
  3. C2n = o(n), because 2n grows slowly
  4. Dn = o(n), since every function is little-o of itself
3.Data structuresMedium

Why is traversing an array often much faster in practice than traversing a linked list of the same length?

  1. ALinked lists always store elements on disk
  2. BArray elements are contiguous, so they make good use of the CPU cache
  3. CArrays use fewer instructions to compare values
  4. DArray traversal is O(1) in total, while linked list traversal is O(n)
4.Data structuresHard

Under simple uniform hashing, what is the expected time of a search in a hash table with chaining, n keys and m buckets?

  1. AO(1 + α), where α = n / m
  2. BO(n)
  3. CO(m)
  4. DO(log n)
5.Graph theoryMedium

A tree has 4 vertices of degree 4, and all its other vertices are leaves (degree 1). How many leaves L does it have?

4 × deg 4
L =
6.Graph theoryHard
109815211415ABCDEF

Use Dijkstra’s algorithm to find the length d(F) of the shortest path from A to F in the weighted graph.

start: Aend: F
d(F) =

3Solutions

1.Algorithms & complexity

Answer: B · 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

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

Answer: B · n = o(n²), but n is not o(n)

Why this is the answer

f = o(g) means f(n) / g(n) tends to 0; n / n² tends to 0, but n / n = 1 does not.

Answer

B · n = o(n²), but n is not o(n)
3.Data structures

Answer: B · Array elements are contiguous, so they make good use of the CPU cache

Why this is the answer

Contiguous memory gives good locality of reference, while linked nodes may be scattered and cause many cache misses.

Answer

B · Array elements are contiguous, so they make good use of the CPU cache
4.Data structures

Answer: A · O(1 + α), where α = n / m

Why this is the answer

Computing the hash costs O(1) and the expected chain length is α, so keeping α bounded by resizing gives O(1) expected time.

Answer

A · O(1 + α), where α = n / m
5.Graph theory

Answer: L = 10

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.

Handshake lemma for a tree

L + 16 = 2(L + 4) − 2
L = 10

Answer

L = 10

Shortcut: L = 2 + Σ (d − 2) over the vertices with degree d ≥ 3.

6.Graph theory

Answer: d(F) = 15

Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.

Dijkstra round by round

Each line: the vertex settled, its final distance, and the neighbours whose distance improves.

A (0): B: 10, C: 9
C (9): E: 11
B (10): D: 25
E (11): F: 15
F (15)
D (25)

Shortest path

A → C → E → F
d(F) = 15

Answer

d(F) = 15

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