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

What is the amortised time complexity of appending to a dynamic array that doubles its capacity when full?

  1. AO(n²)
  2. BO(log n)
  3. CO(1)
  4. DO(n)
2.Algorithms & complexityHard

Insertion sort runs in O(n) on sorted input. Does this contradict the Ω(n log n) lower bound for comparison sorting?

  1. ANo, the lower bound applies to the worst case over all inputs, not to every input
  2. BNo, because insertion sort is not a comparison sort
  3. CYes, but only for arrays shorter than 100
  4. DYes, the lower bound is therefore wrong
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. BArrays use fewer instructions to compare values
  3. CArray traversal is O(1) in total, while linked list traversal is O(n)
  4. DArray elements are contiguous, so they make good use of the CPU cache
4.Data structuresHard

Kahn's algorithm for topological sorting repeatedly removes vertices with in-degree 0. Which structure holds those vertices?

  1. AA queue (or any collection) of vertices whose in-degree is 0
  2. BAn adjacency matrix recording which vertices have already been removed
  3. CA max-heap ordered by out-degree
  4. DA stack of all visited edges
5.Graph theoryMedium

A connected graph has 10 vertices and 20 edges. How many edges k must be removed to leave a spanning tree?

|V| = 10|E| = 20
k =
6.Graph theoryHard
78175691410ABCDEF

Find the total weight W of a minimum spanning tree of the weighted graph.

|V| = 6|E| = 8
W =

3Solutions

1.Algorithms & complexity

Answer: C · O(1)

Why this is the answer

Copying all elements happens only at powers of two; spread over all appends, the average cost per append is constant.

Answer

C · O(1)
2.Algorithms & complexity

Answer: A · No, the lower bound applies to the worst case over all inputs, not to every input

Why this is the answer

The bound says every comparison sort needs about n log n comparisons on some input of size n; special inputs can be much faster.

Answer

A · No, the lower bound applies to the worst case over all inputs, not to every input
3.Data structures

Answer: D · 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

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

Answer: A · A queue (or any collection) of vertices whose in-degree is 0

Why this is the answer

Ready vertices are kept in a queue; removing one lowers its neighbours' in-degrees, and any that reach 0 are added, giving O(V + E) time.

Answer

A · A queue (or any collection) of vertices whose in-degree is 0
5.Graph theory

Answer: k = 11

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

10 − 1 = 9

Edges to remove

k = 20 − 9 = 11

Answer

k = 11
6.Graph theory

Answer: W = 40

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.

Kruskal’s algorithm

BD (5): take
CE (6): take
AB (7): take
AC (8): take
DE (9): skip, closes a cycle
BE (10): skip, closes a cycle
EF (14): take

Total weight

W = 5 + 6 + 7 + 8 + 14 = 40

Answer

W = 40

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