Global shortest-path greedy is within a factor of four

For every natural number $N$, every grid network $G$ given by an injective placement $x_i\in\mathbb Z^2$ of $V=\{0,\ldots,N-1\}$, every pair of distinct vertices $a,b\in V$, and every pair of real numbers $p,q$ satisfying $0\le p\le1$ and $0\le q\le1$, the following two inequalities hold:

Here $E_G$ consists of pairs $(i,j)$ with $i<j$ whose positions have Manhattan distance $1$. An encoded route $r$ consists of a length $1\le m(r)\le N^2$ and a vertex sequence $(v_0,\ldots,v_{m(r)})$; it is admissible in an edge set $J$ from $a$ to $b$ when its endpoints are $a,b$, its canonical pairs $(\min(v_i,v_{i+1}),\max(v_i,v_{i+1}))$ are distinct members of $J$, and no intermediate vertex equals either endpoint. Other vertex repetitions are permitted. A feasible family is a finite set of such routes whose edge sets are disjoint for every two distinct members, $Y_q(F)=\sum_{r\in F}q^{m(r)-1}$, and $M_q(H)$ is the maximum of $0$ and all values $Y_q(F)$ for feasible families in $H$. The family $F_H$ is obtained by starting with $H$ and $|H|$ available steps: if there are no steps or no admissible routes, return the empty set; otherwise select a minimum-length admissible route by a fixed classical choice, delete its edge set, recursively continue with one fewer step, and insert that route into the resulting finite set. The outer sums include all subsets of $E_G$, including the empty subset; empty family sums are $0$, and every zeroth power, including $0^0$, equals $1$. Thus the stated parameter ranges include $p=0$, $p=1$, $q=0$, and $q=1$. For $N=0$ or $N=1$, there are no distinct endpoint choices satisfying the hypotheses.