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

How much stack space does a simple recursive factorial function fact(n) = n × fact(n − 1) use?

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

How do n! and 2ⁿ compare for large n?

  1. An! is O(n²)
  2. B2ⁿ grows faster than n!
  3. CThey grow at the same rate
  4. Dn! grows faster than 2ⁿ
3.Data structuresMedium

Which data structure does a running program use to keep track of active function calls and their local variables?

  1. AThe heap memory queue
  2. BA binary search tree of frames
  3. CThe call stack
  4. DA hash table of functions
4.Data structuresHard

What is the main advantage of a skip list over a sorted linked list?

  1. AIt stores keys without order
  2. BGuaranteed O(1) search because every node stores a pointer to every other node
  3. CIt uses no extra memory
  4. DExpected O(log n) search by using several levels of forward pointers
5.Graph theoryMedium

A connected graph has 20 vertices and 28 edges. How many edges k must be removed to leave a spanning tree?

|V| = 20|E| = 28
k =
6.Graph theoryHard
5123171132011ABCDEF

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

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

3Solutions

1.Algorithms & complexity

Answer: D · O(n)

Why this is the answer

The recursion is n calls deep before any returns, and each call needs a stack frame, so it uses O(n) space.

Answer

D · O(n)
2.Algorithms & complexity

Answer: D · n! grows faster than 2ⁿ

Why this is the answer

For n ≥ 4, n! exceeds 2ⁿ, and the ratio grows without bound because each extra factor of n! is larger than 2.

Answer

D · n! grows faster than 2ⁿ
3.Data structures

Answer: C · The call stack

Why this is the answer

Each call pushes a stack frame and each return pops it, which is why the most recent call finishes first.

Answer

C · The call stack
4.Data structures

Answer: D · Expected O(log n) search by using several levels of forward pointers

Why this is the answer

Randomised levels allow searches to skip many nodes, giving expected O(log n) search, insert and delete.

Answer

D · Expected O(log n) search by using several levels of forward pointers
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

20 − 1 = 19

Edges to remove

k = 28 − 19 = 9

Answer

k = 9
6.Graph theory

Answer: W = 42

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

CE (1): take
BC (3): take
AB (5): take
BE (11): skip, closes a cycle
AC (12): skip, closes a cycle
DE (13): take
BD (17): skip, closes a cycle
DF (20): take

Total weight

W = 1 + 3 + 5 + 13 + 20 = 42

Answer

W = 42

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