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(n²)
  2. BO(log n)
  3. CO(n)
  4. DO(1)
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. Cn = o(n²), but n is not o(n)
  4. Dn² = o(n)
3.Data structuresMedium

A hash table of size 7 uses h(k) = k mod 7 and linear probing. After inserting 10, where is 17 stored?

  1. AIndex 3
  2. BIndex 6
  3. CIndex 0
  4. DIndex 4
4.Data structuresHard

Which statement about heapsort is correct?

  1. AIt needs a binary search tree as extra storage, so it uses O(n) extra memory
  2. BIt sorts in place in O(n log n) worst-case time but is not stable
  3. CIt is stable and needs O(n) extra memory
  4. DIt runs in O(n) time on any input
5.Graph theoryMedium
ABCDEEuler circuitEuler path, no circuitneither

Does the connected graph have an Euler circuit, an Euler path that is not a circuit, or neither? Click your answer.

|V| = 5|E| = 5
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
1771851510681ABCDEF

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

|V| = 6|E| = 9
W =

3Solutions

1.Algorithms & complexity

Answer: C · 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

C · O(n)
2.Algorithms & complexity

Answer: C · 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

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

Answer: D · Index 4

Why this is the answer

10 mod 7 = 3 and 17 mod 7 = 3; index 3 is taken, so linear probing tries the next slot, index 4.

Answer

D · Index 4
4.Data structures

Answer: B · It sorts in place in O(n log n) worst-case time but is not stable

Why this is the answer

Heapsort builds a max-heap and extracts the maximum n times within the same array; swaps over long distances can reorder equal keys.

Answer

B · It sorts in place in O(n log n) worst-case time but is not stable
5.Graph theory

Answer: neither

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.

Degrees

deg(A) = 1, deg(B) = 2, deg(C) = 3, deg(D) = 1, deg(E) = 3

Odd vertices

A, C, D, E → 4
neither

Answer

neither
6.Graph theory

Answer: W = 34

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

BE (1): take
BD (5): take
DF (6): take
AC (7): take
EF (8): skip, closes a cycle
DE (10): skip, closes a cycle
CE (15): take

Total weight

W = 1 + 5 + 6 + 7 + 15 = 34

Answer

W = 34

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