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 worst-case time complexity of quicksort?

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

Which statement about little-o notation is correct?

  1. A2n = o(n), because 2n grows slowly
  2. Bn = o(n), since every function is little-o of itself
  3. Cn² = o(n)
  4. Dn = o(n²), but n is not o(n)
3.Data structuresMedium

A trie (prefix tree) is especially efficient for:

  1. ASorting numbers
  2. BPrefix searches such as autocomplete
  3. CStoring matrices
  4. DFinding shortest paths
4.Data structuresHard

How does a radix tree (compressed trie) improve on a plain trie?

  1. AIt sorts the children of each node by frequency
  2. BIt keeps only the first letter of each word
  3. CIt merges chains of single-child nodes into one edge labelled with a string
  4. DIt stores each word in a hash table at the root instead of in separate nodes
5.Graph theoryMedium

A connected graph has 6 vertices and 8 edges. How many edges k must be removed to leave a spanning tree?

|V| = 6|E| = 8
k =
6.Graph theoryHard
7161814189124ABCDEFG

Kruskal’s algorithm builds a minimum spanning tree. Click the edge it adds as edge number 6; skipped edges do not count.

k = 6
  1. 1edge AB
  2. 2edge AC
  3. 3edge BD
  4. 4edge CE
  5. 5edge DE
  6. 6edge EF
  7. 7edge GB
  8. 8edge GD
  9. 9edge GE

3Solutions

1.Algorithms & complexity

Answer: C · O(n²)

Why this is the answer

With consistently bad pivots (e.g. already sorted input and first-element pivot) partitions are unbalanced, giving O(n²). The average case is O(n log n).

Answer

C · O(n²)
2.Algorithms & complexity

Answer: D · 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

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

Answer: B · 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

B · Prefix searches such as autocomplete
4.Data structures

Answer: C · It merges chains of single-child nodes into one edge labelled with a string

Why this is the answer

Collapsing paths with no branching reduces the number of nodes and the memory used while keeping prefix search.

Answer

C · It merges chains of single-child nodes into one edge labelled with a string
5.Graph theory

Answer: k = 3

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

6 − 1 = 5

Edges to remove

k = 8 − 5 = 3

Answer

k = 3
6.Graph theory

Answer: edge EF

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

1. BD (1)
2. GE (4)
3. AB (7)
4. CE (8)
5. GB (9)
6. EF (18)

edge number 6

EF

Answer

edge EF

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