Finding a Happy Clique

Theorem 11.14   The oriented neighborhood of every oriented vertex contains a happy clique.

Let $v_1,\ldots,v_k$ be the oriented vertices in $OV(\pi )$ in increasing left endpoint order (we ignore unoriented vertices in this stage). To locate a happy clique in $OV(\pi )$, The algorithm traverses the oriented vertices in $OV(\pi )$ according to this order. Let L(e) and R(e) be the left and right endpoints, respectively, of the interval corresponding to a vertex e in the realization of $OV(\pi )$. After traversing $v_1,\ldots,v_i$ for $1\le i\le k$, the algorithm maintains a happy clique C_i in the subgraph of $OV(\pi )$ induced by these vertices. Assume |C_i|=j, $j\le i$ and let $e_{i_1},\ldots,e_{i_j}$ be the vertices in C_i where $i_1 < i_2 <\ldots < i_j$. The vertices of C_i are maintained in a linked list ordered in increasing left endpoint order. If there exists an interval that contains all the intervals in C_i then the algorithm maintains a minimal such interval t_i. The clique C_i and the vertex t_i (if exists) satisfy the following invariant:

Invariant 11.4.1   $\;$
1) Every vertex $v_l\not\in C_i$, $l \le i$, such that L(v_i1) < L(v_l) must be adjacent to t_i.
2) Every vertex $v_l\not\in C_i$, L(v_l) < L(v_i1) that is adjacent to a vertex in C_i is either adjacent to an interval v_p such that R(v_p) < L(v_i1) or adjacent to t_i.

We prove the correctness of this invariant by induction: Initially $C_1=\{v_1\}$ and t₁ is undefined. If R(e_ij) < L(e_i+1) then C_i is guaranteed to be happy in $OV(\pi )$ (see figure 
reflec11:Fig:fighappy(a)). We need to focus only on cases with $L(e_{i+1}) \le R(e_{i_j})$. The induction step: We assume correctness up until i and show how to obtain C_i+1 and t_i+1 if $L(e_{i+1}) \le R(e_{i_j})$. We have to consider the following cases:
Case 1. The interval t_i is defined and R(t_i) <R(v_i+1). Continue with C_i+1 = C_i and t_i+1 = t_i. See figure 11.11(b).
Case 2. The interval t_i is not defined or $R(v_{i+1}) \le R(t_i)$.
a) R(v_ij) < R(v_i+1) and $L(v_{i+1}) \le R(v_{i_1})$. C_i+1 is obtained by adding v_i+1 to C_i and t_i+1 = t_i. See figure 11.11(c).
b) R(v_ij) < R(v_i+1) and L(v_i+1) > R(v_i1). The clique C_i+1 consists of v_i+1 alone and t_i+1 = t_i. See figure 11.11(d).
c) R(v_i+1) < R(v_ij). As in the previous case $C_{i+1} = \{ v_{i+1} \}$. In this case t_i+1 is set to v_ij, the last interval in C_i. See figure 11.11(e).

  

Figure 11.11: The various cases of the algorithm for finding a happy clique. The topmost interval is always t_i. The three thick intervals comprise C_i. The dotted interval corresponds to v_i+1.

The fact that C_i is happy in the subgraph induced by $v_1,\ldots,v_i$ follows from this invariant. It is straightforward to see that the clique C_l that the algorithm stops with, is happy. The running time of the algorithm is proportional to the number of oriented vertices traversed since a constant amount of work is performed for each such vertex.